GEOMETRY AND DYNAMICS IN GROMOV HYPERBOLIC
METRIC SPACES

arXiv:1409.2155v7 [math.DS] 28 Jun 2016

WITH AN EMPHASIS ON NON-PROPER SETTINGS

Tushar Das
David Simmons
Mariusz Urbański
University of Wisconsin – La Crosse, Department of Mathematics
& Statistics, 1725 State Street, La Crosse, WI 54601, USA
E-mail address: tdas@uwlax.edu
URL: https://sites.google.com/a/uwlax.edu/tdas/
University of York, Department of Mathematics, Heslington, York
YO10 5DD, UK
E-mail address: David.Simmons@york.ac.uk
URL: https://sites.google.com/site/davidsimmonsmath/
University of North Texas, Department of Mathematics, 1155 Union
Circle #311430, Denton, TX 76203-5017, USA
E-mail address: urbanski@unt.edu
URL: http://www.urbanskimath.com/

2010 Mathematics Subject Classification. Primary 20H10, 28A78, 37F35, 20F67,
20E08; Secondary 37A45, 22E65, 20M20
Key words and phrases. hyperbolic geometry, Gromov hyperbolic metric space,
infinite-dimensional symmetric space, metric tree, Hausdorff dimension, Poincaré
exponent, Patterson–Sullivan measure, conformal measure, divergence type,
geometrically finite group, global measure formula, exact dimensional measure
Abstract. Our monograph presents the foundations of the theory of groups
and semigroups acting isometrically on Gromov hyperbolic metric spaces. We
make it a point to avoid any assumption of properness/compactness, keeping in mind the motivating example of H∞ , the infinite-dimensional rank-one
symmetric space of noncompact type over the reals. We have not skipped
over parts that might be thought of as “trivial” extensions of the finitedimensional/proper theory, as our intuition has turned out to be wrong often
enough regarding these matters that we feel it is worth writing everything down
explicitly and with an eye to detail. Moreover, we feel that it is a methodological advantage to have the entire theory presented from scratch, in order
to provide a basic reference for the theory.
Though our work unifies and extends a long list of results obtained by
many authors, Part 1 of this monograph may be treated as mostly expository.
The remainder of this work, some of whose highlights are described in brief
below, contains several new methods, examples, and theorems. In Part 2, we
introduce a modification of the Poincaré exponent, an invariant of a group
which provides more information than the usual Poincaré exponent, which we
then use to vastly generalize the Bishop–Jones theorem relating the Hausdorff
dimension of the radial limit set to the Poincaré exponent of the underlying
semigroup. We construct examples which illustrate the surprising connection
between Hausdorff dimension and various notions of discreteness which show
up in non-proper settings. Part 3 of the monograph provides a number of
examples of groups acting on H∞ which exhibit a wide range of phenomena not
to be found in the finite-dimensional theory. Such examples often demonstrate
the optimality of our theorems.
In Part 4, we construct Patterson–Sullivan measures for groups of divergence type without any compactness assumption on either the boundary
or the limit set. This is carried out by first constructing such measures on
the Samuel–Smirnov compactification of the bordification of the underlying
hyperbolic space, and then showing that the measures are supported on the
(non-compactified) bordification. We end with a study of quasiconformal measures of geometrically finite groups in terms of doubling and exact dimensionality. Our analysis characterizes exact dimensionality in terms of Diophantine
approximation on the boundary. We demonstrate that though all doubling
Patterson–Sullivan measures are exact dimensional, there exist Patterson–
Sullivan measures that are exact dimensional but not doubling, as well as
ones that are neither doubling nor exact dimensional.

Dedicated to the memory of our friend

Bernd O. Stratmann
Mathematiker
17th July 1957 – 8th August 2015

Contents
List of Figures

xiii

Prologue

xv

Chapter 1. Introduction and Overview
1.1. Preliminaries

xix
xx

1.1.1. Algebraic hyperbolic spaces
1.1.2. Gromov hyperbolic metric spaces
1.1.3. Discreteness
1.1.4. The classification of semigroups
1.1.5. Limit sets
1.2. The Bishop–Jones theorem and its generalization
1.2.1. The modified Poincaré exponent
1.3. Examples
1.3.1. Schottky products

xx
xxi
xxv
xxvi
xxvii
xxviii
xxxi
xxxii
xxxiii

1.3.2. Parabolic groups
1.3.3. Geometrically finite and convex-cobounded groups

xxxiv
xxxiv

1.3.4. Counterexamples
1.3.5. R-trees and their isometry groups

xxxv
xxxvi

1.4. Patterson–Sullivan theory
1.4.1. Quasiconformal measures of geometrically finite groups
1.5. Appendices
Part 1.

xxxvi
xxxix
xli

Preliminaries

1

Chapter 2. Algebraic hyperbolic spaces
2.1. The definition
2.2. The hyperboloid model
2.3. Isometries of algebraic hyperbolic spaces

3
3
4
8

2.4. Totally geodesic subsets of algebraic hyperbolic spaces

14

2.5. Other models of hyperbolic geometry
2.5.1. The (Klein) ball model

18
18

2.5.2. The half-space model

19
vii

viii

CONTENTS

2.5.3. Transitivity of the action of Isom(H) on ∂H
Chapter 3. R-trees, CAT(-1) spaces, and Gromov hyperbolic metric spaces
3.1. Graphs and R-trees

21
23
23

3.2. CAT(-1) spaces
3.2.1. Examples of CAT(-1) spaces

27
28

3.3. Gromov hyperbolic metric spaces
3.3.1. Examples of Gromov hyperbolic metric spaces

29
31

3.4. The boundary of a hyperbolic metric space
3.4.1. Extending the Gromov product to the boundary

33
35

3.4.2. A topology on bord X

38

3.5. The Gromov product in algebraic hyperbolic spaces
3.5.1. The Gromov boundary of an algebraic hyperbolic space

41
46

3.6. Metrics and metametrics on bord X
3.6.1. General theory of metametrics

48
48

3.6.2. The visual metametric based at a point w ∈ X
3.6.3. The extended visual metric on bord X

50
51

3.6.4. The visual metametric based at a point ξ ∈ ∂X

53

Chapter 4. More about the geometry of hyperbolic metric spaces

59

4.1. Gromov triples
4.2. Derivatives

59
60

4.2.1. Derivatives of metametrics
4.2.2. Derivatives of maps

60
61

4.2.3. The dynamical derivative

63

4.3. The Rips condition
4.4. Geodesics in CAT(-1) spaces

65
67

4.5. The geometry of shadows
4.5.1. Shadows in regularly geodesic hyperbolic metric spaces

73
73

4.5.2. Shadows in hyperbolic metric spaces
4.6. Generalized polar coordinates

74
79

Chapter 5. Discreteness
5.1. Topologies on Isom(X)
5.2. Discrete groups of isometries
5.2.1. Topological discreteness

83
83
87
89

5.2.2. Equivalence in finite dimensions

91

5.2.3. Proper discontinuity
5.2.4. Behavior with respect to restrictions

92
94

5.2.5. Countability of discrete groups

94

CONTENTS

ix

Chapter 6. Classification of isometries and semigroups

95

6.1. Classification of isometries
6.1.1. More on loxodromic isometries

95
97

6.1.2. The story for real hyperbolic spaces
6.2. Classification of semigroups

98
99

6.2.1. Elliptic semigroups
6.2.2. Parabolic semigroups

100
100

6.2.3. Loxodromic semigroups
6.3. Proof of the Classification Theorem

101
103

6.4. Discreteness and focal groups

105

Chapter 7. Limit sets
7.1. Modes of convergence to the boundary

109
109

7.2. Limit sets
7.3. Cardinality of the limit set

112
113

7.4. Minimality of the limit set

114

7.5. Convex hulls
7.6. Semigroups which act irreducibly on algebraic hyperbolic spaces

119
122

7.7. Semigroups of compact type

123

Part 2.

The Bishop–Jones theorem

127

Chapter 8. The modified Poincaré exponent
8.1. The Poincaré exponent of a semigroup

129
129

8.2. The modified Poincaré exponent of a semigroup

130

Chapter 9. Generalization of the Bishop–Jones theorem
9.1. Partition structures

135
136

9.2. A partition structure on ∂X
9.3. Sufficient conditions for Poincaré regularity
Part 3.

141
148

Examples

155

Chapter 10. Schottky products

157

10.1. Free products
10.2. Schottky products

157
158

10.3. Strongly separated Schottky products

160

10.4. A partition-structure–like structure
10.5. Existence of Schottky products

168
171

Chapter 11. Parabolic groups
11.1. Examples of parabolic groups acting on E

175
∞

175

x

CONTENTS

11.1.1. The Haagerup property and the absence of a Margulis lemma
11.1.2. Edelstein examples
11.2. The Poincaré exponent of a finitely generated parabolic group

176
177
183

11.2.1. Nilpotent and virtually nilpotent groups
11.2.2. A universal lower bound on the Poincaré exponent

183
185

11.2.3. Examples with explicit Poincaré exponents

186

Chapter 12. Geometrically finite and convex-cobounded groups
12.1. Some geometric shapes
12.1.1. Horoballs
12.1.2. Dirichlet domains

193
193
193
194

12.2. Cobounded and convex-cobounded groups
12.2.1. Characterizations of convex-coboundedness

196
198

12.2.2. Consequences of convex-coboundedness
12.3. Bounded parabolic points

201
201

12.4. Geometrically finite groups
12.4.1. Characterizations of geometrical finiteness

206
207

12.4.2. Consequences of geometrical finiteness

213

12.4.3. Examples of geometrically finite groups

218

Chapter 13. Counterexamples

221

13.1. Embedding R-trees into real hyperbolic spaces
13.2. Strongly discrete groups with infinite Poincaré exponent

222
226

13.3. Moderately discrete groups which are not strongly discrete

226

13.4. Poincaré irregular groups
13.5. Miscellaneous counterexamples

228
232

Chapter 14. R-trees and their isometry groups
14.1. Construction of R-trees by the cone method

233
233

14.2. Graphs with contractible cycles

237

14.3. The nearest-neighbor projection onto a convex set
14.4. Constructing R-trees by the stapling method

239
240

14.5. Examples of R-trees constructed using the stapling method

245

Part 4.

Patterson–Sullivan theory

Chapter 15. Conformal and quasiconformal measures

253
255

15.1. The definition

255

15.2. Conformal measures
15.3. Ergodic decomposition

256
256

15.4. Quasiconformal measures

259

CONTENTS

xi

15.4.1. Pointmass quasiconformal measures

260

15.4.2. Non-pointmass quasiconformal measures

261

Chapter 16. Patterson–Sullivan theorem for groups of divergence type
16.1. Samuel–Smirnov compactifications
b
16.2. Extending the geometric functions to X
b
16.3. Quasiconformal measures on X

267
267
268
270

16.4. The main argument
16.5. End of the argument

273
276

16.6. Necessity of the generalized divergence type assumption
16.7. Orbital counting functions of nonelementary groups

277
279

Chapter 17. Quasiconformal measures of geometrically finite groups

281

17.1. Sufficient conditions for divergence type
17.2. The global measure formula

281
284

17.3. Proof of the global measure formula
17.4. Groups for which µ is doubling

289
295

17.5. Exact dimensionality of µ
17.5.1. Diophantine approximation on Λ

303
304

17.5.2. Examples and non-examples of exact dimensional measures

308

Appendix A. Open problems

313

Appendix B. Index of defined terms

315

Bibliography

321

List of Figures
3.1.1 A geodesic triangle in an R-tree

27

3.3.1 A quadruple of points in an R-tree

30

3.3.2 Expressing distance via Gromov products in an R-tree

32

3.4.1 A Gromov sequence in an R-tree

34

3.5.1 Relating angle and the Gromov product

42

3.5.2 B is strongly Gromov hyperbolic

45

3.5.3 A formula for the Busemann function in the half-space model

48

3.6.1 The Hamenstädt distance

56

4.2.1 The derivative of g at ∞

63

4.3.1 The Rips condition

66

4.4.1 The triangle ∆(x, y1 , y2 )

69

4.5.1 Shadows in regularly geodesic hyperbolic metric spaces

74

4.5.2 The Intersecting Shadows Lemma

76

4.5.3 The Big Shadows Lemma

77

4.5.4 The Diameter of Shadows Lemma

78

4.6.1 Polar coordinates in the half-space model

79

6.4.1 High altitude implies small displacement in the half-space model

107

7.1.1 Conical convergence to the boundary

110

7.1.2 Converging horospherically but not radially to the boundary

111

9.2.1 The construction of children

142

9.2.2 The sets Cn , for n ∈ Z

143

10.3.1 The strong separation lemma for Schottky products

162

12.1.1 Visualizing horoballs in the ball and half-space models

194

12.1.2 Diameter decay of a ball complement inside a horoball

195

xiii

xiv

LIST OF FIGURES

12.1.3 The Cayley graph of Γ = F2 (Z) = hγ1 , γ2 i

196

12.2.1 Proving that convex-cobounded groups are of compact type

200

12.3.1 The geometry of bounded parabolic points

205

12.4.1 Proving that geometrically finite groups are of compact type

209

12.4.2 Local finiteness of the horoball collection

211

12.4.3 Orbit maps of geometrically finite groups are QI embeddings

215

13.4.1 Geometry of automorphisms of a simplicial tree

229

14.2.1 Triangles in graphs with contractible cycles

239

14.2.2 Triangles in graphs with contractible cycles: general case

239

14.4.1 The consistency condition for stapling metric spaces

242

14.5.1 The Cayley graph of F2 (Z) as a pure Schottky product

248

14.5.2 An example of a geometric product

251

14.5.3 Another example of a geometric product

252

17.2.1 Cusp excursion and ball measure functions

285

17.3.1 Estimating measures of balls via information “at infinity”

290

17.3.2 Estimating measures of balls via “local” information

293

Prologue
. . . Cela suffit pour faire comprendre que dans les cinq mémoires
des Acta mathematica que j’ai consacrés à l’étude des transcendantes fuchsiennes et kleinéennes, je n’ai fait qu’effleurer un sujet très vaste, qui fournira sans doute aux géomètres l’occasion de
nombreuses et importantes découvertes.1
– H. Poincaré, Acta Mathematica, 5, 1884, p. 278.
The theory of discrete subgroups of real hyperbolic space has a long history. It
was inaugurated by Poincaré, who developed the two-dimensional (Fuchsian) and
three-dimensional (Kleinian) cases of this theory in a series of articles published
between 1881 and 1884 that included numerous notes submitted to the C. R. Acad.
Sci. Paris, a paper at Klein’s request in Math. Annalen, and five memoirs commissioned by Mittag-Leffler for his then freshly-minted Acta Mathematica. One
must also mention the complementary work of the German school that came before Poincaré and continued well after he had moved on to other areas, viz. that
of Klein, Schottky, Schwarz, and Fricke. See [80, Chapter 3] for a brief exposition of this fascinating history, and [79, 63] for more in-depth presentations of the
mathematics involved.
We note that in finite dimensions, the theory of higher-dimensional Kleinian
groups, i.e., discrete isometry groups of the hyperbolic d-space Hd for d ≥ 4, is
markedly different from that in H3 and H2 . For example, the Teichmüller theory used by the Ahlfors–Bers school (viz. Marden, Maskit, Jørgensen, Sullivan,
Thurston, etc.) to study three-dimensional Kleinian groups has no generalization
to higher dimensions. Moreover, the recent resolution of the Ahlfors measure conjecture [3, 43] has more to do with three-dimensional topology than with analysis
and dynamics. Indeed, the conjecture remains open in higher dimensions [106, p.
526, last paragraph]. Throughout the twentieth century, there are several instances
of theorems proven for three-dimensional Kleinian groups whose proofs extended
1This is enough to make it apparent that in these five memoirs in Acta Mathematica which I have
dedicated to the study of Fuschian and Kleinian transcendants, I have only skimmed the surface
of a very broad subject, which will no doubt provide geometers with the opportunity for many
important discoveries.
xv

xvi

PROLOGUE

easily to n dimensions (e.g. [21, 133]), but it seems that the theory of higherdimensional Kleinian groups was not really considered a subject in its own right
until around the 1990s. For more information on the theory of higher-dimensional
Kleinian groups, see the survey article [106], which describes the state of the art
up to the last decade, emphasizing connections with homological algebra.
But why stop at finite n? Dennis Sullivan, in his IHÉS Seminar on Conformal
and Hyperbolic Geometry [164] that ran during the late 1970s and early ’80s, indicated a possibility of developing the theory of discrete groups acting by hyperbolic
isometries on the open unit ball of a separable infinite-dimensional Hilbert space.2
Later in the early ’90s, Misha Gromov observed the paucity of results regarding
such actions in his seminal lectures Asymptotic Invariants of Infinite Groups [86]
where he encouraged their investigation in memorable terms: “The spaces like this
[infinite-dimensional symmetric spaces] . . . look as cute and sexy to me as their
finite dimensional siblings but they have been for years shamefully neglected by
geometers and algebraists alike”.
Gromov’s lament had not fallen to deaf ears, and the geometry and representation theory of infinite-dimensional hyperbolic space H∞ and its isometry group
have been studied in the last decade by a handful of mathematicians, see e.g.
[40, 65, 132]. However, infinite-dimensional hyperbolic geometry has come into
prominence most spectacularly through the recent resolution of a long-standing
conjecture in algebraic geometry due to Enriques from the late nineteenth century. Cantat and Lamy [47] proved that the Cremona group (i.e. the group of
birational transformations of the complex projective plane) has uncountably many
non-isomorphic normal subgroups, thus disproving Enriques’ conjecture. Key to
their enterprise is the fact, due to Manin [125], that the Cremona group admits a
faithful isometric action on a non-separable infinite-dimensional hyperbolic space,
now known as the Picard–Manin space.
Our project was motivated by a desire to answer Gromov’s plea by exposing a
coherent general theory of groups acting isometrically on the infinite-dimensional
hyperbolic space H∞ . In the process we came to realize that a more natural domain for our inquiries was the much larger setting of semigroups acting on Gromov hyperbolic metric spaces – that way we could simultaneously answer our own
questions about H∞ and construct a theoretical framework for those who are interested in more exotic spaces such as the curve graph, arc graph, and arc complex
[95, 126, 96] and the free splitting and free factor complexes [89, 27, 104, 96].
2This was the earliest instance of such a proposal that we could find in the literature, although

(as pointed out to us by P. de la Harpe) infinite-dimensional hyperbolic spaces without groups
acting on them had been discussed earlier [130, §27], [131, 60]. It would be of interest to know
whether such an idea may have been discussed prior to that.

PROLOGUE

xvii

These examples are particularly interesting as they extend the well-known dictionary [26, p.375] between mapping class groups and the groups Out(FN ). In
another direction, a dictionary is emerging between mapping class groups and Cremona groups, see [30, 66]. We speculate that developing the Patterson–Sullivan
theory in these three areas would be fruitful and may lead to new connections and
analogies that have not surfaced till now.
In a similar spirit, we believe there is a longer story for which this monograph
lays the foundations. In general, infinite-dimensional space is a wellspring of outlandish examples and the wide range of new phenomena we have started to uncover
has no analogue in finite dimensions. The geometry and analysis of such groups
should pique the interests of specialists in probability, geometric group theory, and
metric geometry. More speculatively, our work should interact with the ongoing
and still nascent study of geometry, topology, and dynamics in a variety of infinitedimensional spaces and groups, especially in scenarios with sufficient negative curvature. Here are three concrete settings that would be interesting to consider: the
universal Teichmüller space, the group of volume-preserving diffeomorphisms of R3
or a 3-torus, and the space of Kähler metrics/potentials on a closed complex manifold in a fixed cohomology class equipped with the Mabuchi–Semmes–Donaldson
metric. We have been developing a few such themes. The study of thermodynamics
(equilibrium states and Gibbs measures) on the boundaries of Gromov hyperbolic
spaces will be investigated in future work [57]. We speculate that the study of
stochastic processes (random walks and Brownian motion) in such settings would
be fruitful. Furthermore, it would be of interest to develop the theory of discrete
isometric actions and limit sets in infinite-dimensional spaces of higher rank.
Acknowledgements. This monograph is dedicated to our colleague Bernd O.
Stratmann, who passed away on the 8th of August, 2015. Various discussions with
Bernd provided inspiration for this project and we remain grateful for his friendship.
The authors thank D. P. Sullivan, D. Mumford, B. Farb, P. Pansu, F. Ledrappier,
A. Wilkinson, K. Biswas, E. Breuillard, A. Karlsson, I. Assani, M. Lapidus, R.
Guo, Z. Huang, I. Gekhtman, G. Tiozzo, P. Py, M. Davis, M. Roychowdury, M.
Hochman, J. Tao, P. de la Harpe, T. Barthelme, J. P. Conze, and Y. Guivarc’h
for their interest and encouragement, as well as for invitations to speak about our
work at various venues. We are grateful to S. J. Patterson, J. Elstrodt, and É. Ghys
for enlightening historical discussions on various themes relating to the history of
Fuchsian and Kleinian groups and their study through the twentieth century, and to
D. P. Sullivan and D. Mumford for suggesting work on diffeomorphism groups and
the universal Teichmüller space. We thank X. Xie for pointing us to a solution to
one of the problems in our Appendix A. The research of the first-named author was

xviii

PROLOGUE

supported in part by 2014-2015 and 2016-2017 Faculty Research Grants from the
University of Wisconsin–La Crosse. The research of the second-named author was
supported in part by the EPSRC Programme Grant EP/J018260/1. The research
of the third-named author was supported in part by the NSF grant DMS-1361677.

CHAPTER 1

Introduction and Overview
The purpose of this monograph is to present the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces in full detail as
we understand it, with special emphasis on the case of infinite-dimensional algebraic hyperbolic spaces X = H∞
F , where F denotes a division algebra. We have
not skipped over the parts which some would call “trivial” extensions of the finitedimensional/proper theory, for two main reasons: first, intuition has turned out to
be wrong often enough regarding these matters that we feel it is worth writing everything down explicitly; second, we feel it is better methodologically to present the
entire theory from scratch, in order to provide a basic reference for the theory, since
no such reference exists currently (the closest, [39], has a fairly different emphasis).
Thus Part 1 of this monograph should be treated as mostly expository, while Parts
2-4 contain a range of new material. For experts who want a geodesic path to
significant theorems, we list here five such results that we prove in this monograph:
Theorems 1.2.1 and 1.4.4 provide generalizations of the Bishop–Jones theorem [28,
Theorem 1] and the Global Measure Formula [160, Theorem 2], respectively, to
Gromov hyperbolic metric spaces. Theorem 1.4.1 guarantees the existence of a
δ-quasiconformal measure for groups of divergence type, even if the space they are
acting on is not proper. Theorem 1.4.5 provides a sufficient condition for the exact
dimensionality of the Patterson-Sullivan measure of a geometrically finite group,
and Theorem 1.4.6 relates the exact dimensionality to Diophantine properties of
the measure. However, the reader should be aware that a sharp focus on just these
results, without care for their motivation or the larger context in which they are situated, will necessarily preclude access to the interesting and uncharted landscapes
that our work has begun to uncover. The remainder of this chapter provides an
overview of these landscapes.
Convention 1. The symbols ., &, and ≍ will denote coarse asymptotics;
a subscript of + indicates that the asymptotic is additive, and a subscript of ×

indicates that it is multiplicative. For example, A .×,K B means that there exists
a constant C > 0 (the implied constant ), depending only on K, such that A ≤ CB.
Moreover, A .+,× B means that there exist constants C1 , C2 > 0 so that A ≤

C1 B + C2 . In general, dependence of the implied constant(s) on universal objects
xix

xx

1. INTRODUCTION AND OVERVIEW

such as the metric space X, the group G, and the distinguished point o ∈ X (cf.

Notation 1.1.5) will be omitted from the notation.

Convention 2. The notation xn −
→ x means that xn → x as n → ∞, while

the notation xn −−→ x means that

n

n,+

x ≍+ lim sup xn ≍+ lim inf xn ,
n→∞

n→∞

and similarly for xn −−→ x.
n,×

Convention 3. The symbol ⊳ is used to indicate the end of a nested proof.
Convention 4. We use the Iverson bracket notation:

1 statement true
[statement] =
0 statement false

Convention 5. Given a distinguished point o ∈ X, we write
kxk = d(o, x) and kgk = kg(o)k.
1.1. Preliminaries

1.1.1. Algebraic hyperbolic spaces. Although we are mostly interested in
this monograph in the real infinite-dimensional hyperbolic space H∞
R , the complex
∞
and quaternionic hyperbolic spaces H∞
C and HQ are also interesting. In finite
dimensions, these spaces constitute (modulo the Cayley hyperbolic plane1) the rank

one symmetric spaces of noncompact type. In the infinite-dimensional case we retain
this terminology by analogy; cf. Remark 2.2.6. For brevity we will refer to a rank
one symmetric space of noncompact type as an algebraic hyperbolic space.
There are several equivalent ways to define algebraic hyperbolic spaces; these
are known as “models” of hyperbolic geometry. We consider here the hyperboloid
model, ball model (Klein’s, not Poincaré’s), and upper half-space model (which
only applies to algebraic hyperbolic spaces defined over the reals, which we will call
α
α
real hyperbolic spaces), which we denote by Hα
F , BF , and E , respectively. Here F
denotes the base field (either R, C, or Q), and α denotes a cardinal number. We

omit the base field when it is R, and denote the exponent by ∞ when it is #(N),
#(N)
so that H∞ = HR
is the unique separable infinite-dimensional real hyperbolic
space.
The main theorem of Chapter 2 is Theorem 2.3.3, which states that any isometry of an algebraic hyperbolic space must be an “algebraic” isometry. The finitedimensional case is given as an exercise in Bridson–Haefliger [39, Exercise II.10.21].
1We omit all discussion of the Cayley hyperbolic plane H2 , as the algebra involved is too exotic
O
for our taste; cf. Remark 2.1.1.

1.1. PRELIMINARIES

xxi

We also describe the relation between totally geodesic subsets of algebraic hyperbolic spaces and fixed point sets of isometries (Theorem 2.4.7), a relation which
will be used throughout the paper.
Remark 1.1.1. Key to the study of finite-dimensional algebraic hyperbolic
spaces is the theory of quasiconformal mappings (e.g., as in Mostow and Pansu’s
rigidity theorems [133, 141]). Unfortunately, it appears to be quite difficult to
generalize this theory to infinite dimensions. For example, it is an open question
[92, p.1335] whether every quasiconformal homeomorphism of Hilbert space is also
quasisymmetric.
1.1.2. Gromov hyperbolic metric spaces. Historically, the first motivation for the theory of negatively curved metric spaces came from differential geometry and the study of negatively curved Riemannian manifolds. The idea was
to describe the most important consequences of negative curvature in terms of the
metric structure of the manifold. This approach was pioneered by Aleksandrov
[6], who discovered for each κ ∈ R an inequality regarding triangles in a metric
space with the property that a Riemannian manifold satisfies this inequality if and
only if its sectional curvature is bounded above by κ, and popularized by Gromov,
who called Aleksandrov’s inequality the “CAT(κ) inequality” as an abbreviation
for “comparison inequality of Alexandrov–Toponogov” [85, p.106].2 A metric space
is called CAT(κ) if the distance between any two points on a geodesic triangle is
smaller than the corresponding distance on the “comparison triangle” in a model
space of constant curvature κ; see Definition 3.2.1.
The second motivation came from geometric group theory, in particular the
study of groups acting on manifolds of negative curvature. For example, Dehn
proved that the word problem is solvable for finitely generated Fuchsian groups [64],
and this was generalized by Cannon to groups acting cocompactly on manifolds of
negative curvature [44]. Gromov attempted to give a geometric characterization of
these groups in terms of their Cayley graphs; he tried many definitions (cf. [83,
§6.4], [84, §4]) before converging to what is now known as Gromov hyperbolicity

in 1987 [85, 1.1, p.89], a notion which has influenced much research. A metric
space is said to be Gromov hyperbolic if it satisfies a certain inequality that we call
Gromov’s inequality; see Definition 3.3.2. A finitely generated group is then said
to be word-hyperbolic if its Cayley graph is Gromov hyperbolic.

2It appears that Bridson and Haefliger may be responsible for promulgating the idea that the C in

CAT refers to E. Cartan [39, p.159]. We were unable to find such an indication in [85], although
Cartan is referenced in connection with some theorems regarding CAT(κ) spaces (as are Riemann
and Hadamard).

xxii

1. INTRODUCTION AND OVERVIEW

The big advantage of Gromov hyperbolicity is its generality. We give some idea
of its scope by providing the following nested list of metric spaces which have been
proven to be Gromov hyperbolic:
• CAT(-1) spaces (Definition 3.2.1)

– Riemannian manifolds (both finite- and infinite-dimensional) with
sectional curvature ≤ −1

∗ Algebraic hyperbolic spaces (Definition 2.2.5)
· Picard–Manin spaces of projective surfaces defined over

algebraically closed fields [125], cf. [46, §3.1]
– R-trees (Definition 3.1.10)
∗ Simplicial trees
· Unweighted simplicial trees

• Cayley metrics (Example 3.1.2) on word-hyperbolic groups
• Green metrics on word-hyperbolic groups [29, Corollary 1.2]

• Quasihyperbolic metrics of uniform domains in Banach spaces [173, Theorem 2.12]
• Arc graphs and curve graphs [95] and arc complexes [126, 96] of finitely
punctured oriented surfaces
• Free splitting complexes [89, 96] and free factor complexes [27, 104, 96]

Remark 1.1.2. Many of the above examples admit natural isometric group
actions:
• The Cremona group acts isometrically on the Picard–Manin space [125],
cf. [46, Theorem 3.3].
• The mapping class group of a finitely punctured oriented surface acts
isometrically on its arc graph, curve graph, and arc complex.
• The outer automorphism group Out(FN ) of the free group on N generators
acts isometrically on the free splitting complex F S(FN ) and the free factor
complex F F (FN ).

Remark 1.1.3. Most of the above examples are examples of non-proper hyperbolic metric spaces. Recall that a metric space is said to be proper if its distance
function x 7→ kxk = d(o, x) is proper, or equivalently if closed balls are compact.
Though much of the existing literature on CAT(-1) and hyperbolic metric spaces
assumes that the spaces in question are proper, it is often not obvious whether this

assumption is really essential. However, since results about proper metric spaces
do not apply to infinite-dimensional algebraic hyperbolic spaces, we avoid the assumption of properness.

1.1. PRELIMINARIES

xxiii

Remark 1.1.4. One of the above examples, namely, Green metrics on wordhyperbolic groups, is a natural class of non-geodesic hyperbolic metric spaces.3
However, Bonk and Schramm proved that all non-geodesic hyperbolic metric spaces
can be isometrically embedded into geodesic hyperbolic metric spaces [31, Theorem
4.1], and the equivariance of their construction was proven by Blachère, Haı̈ssinsky,
and Mathieu [29, Corollary A.10]. Thus, one could view the assumption of geodesicity to be harmless, since most theorems regarding geodesic hyperbolic metric spaces
can be pulled back to non-geodesic hyperbolic metric spaces. However, for the most
part we also avoid the assumption of geodesicity, mostly for methodological reasons
rather than because we are considering any particular non-geodesic hyperbolic metric space. Specifically, we felt that Gromov’s definition of hyperbolicity in metric
spaces is a “deep” definition whose consequences should be explored independently
of such considerations as geodesicity. We do make the assumption of geodesicity in Chapter 12, where it seems necessary in order to prove the main theorems.
(The assumption of geodesicity in Chapter 12 can for the most part be replaced by
the weaker assumption of almost geodesicity [31, p.271], but we felt that such a
presentation would be more technical and less intuitive.)
We now introduce a list of standing assumptions and notations. They apply to
all chapters except for Chapters 2, 3, and 5 (see also §4.1).
Notation 1.1.5. Throughout the introduction,
• X is a Gromov hyperbolic metric space (cf. Definition 3.3.2),

• d denotes the distance function of X,
• ∂X denotes the Gromov boundary of X, and bord X denotes the bordification bord X = X ∪ ∂X (cf. Definition 3.4.2),
• D denotes a visual metric on ∂X with respect to a parameter b > 1 and a

distinguished point o ∈ X (cf. Proposition 3.6.8). By definition, a visual
metric satisfies the asymptotic

(1.1.1)

Db,o (ξ, η) ≍× b−hξ|ηio ,
where h·|·i denotes the Gromov product (cf. (3.3.2)).

• Isom(X) denotes the isometry group of X. Also, G ≤ Isom(X) will mean
that G is a subgroup of Isom(X), while G  Isom(X) will mean that G
is a subsemigroup of Isom(X).

A prime example to have in mind is the special case where X is an infinitedimensional algebraic hyperbolic space, in which case the Gromov boundary ∂X
3Quasihyperbolic metrics on uniform domains in Banach spaces can also fail to be geodesic, but
they are almost geodesic which is almost as good. See e.g. [172] for a study of almost geodesic
hyperbolic metric spaces.

xxiv

1. INTRODUCTION AND OVERVIEW

can be identified with the natural boundary of X (Proposition 3.5.3), and we can
set b = e and get equality in (1.1.1) (Observation 3.6.7).
Another important example of a hyperbolic metric space that we will keep in
our minds is the case of R-trees alluded to above. R-trees are a generalization of
simplicial trees, which in turn are a generalization of unweighted simplicial trees,
also known as “Z-trees” or just “trees”. R-trees are worth studying in the context
of hyperbolic metric spaces for two reasons: first of all, they are “prototype spaces”
in the sense that any finite set in a hyperbolic metric space can be roughly isometrically embedded into an R-tree, with a roughness constant depending only on
the cardinality of the set [77, pp.33-38]; second of all, R-trees can be equivariantly
embedded into infinite-dimensional real hyperbolic space H∞ (Theorem 13.1.1),
meaning that any example of a group acting on an R-tree can be used to construct
an example of the same group acting on H∞ . R-trees are also much simpler to
understand than general hyperbolic metric spaces: for any finite set of points, one
can draw out a list of all possible diagrams, and then the set of distances must be
determined from one of these diagrams (cf. e.g., Figure 3.3.1).
Besides introducing R-trees, CAT(-1) spaces, and hyperbolic metric spaces, the
following things are done in Chapter 3: construction of the Gromov boundary
∂X and analysis of its basic topological properties (Section 3.4), proof that the
Gromov boundary of an algebraic hyperbolic space is equal to its natural boundary
(Proposition 3.5.3), and the construction of various metrics and metametrics on the
boundary of X (Section 3.6). None of this is new, although the idea of a metametric
(due to Väisälä [172, §4]) is not very well known.
In Chapter 4, we go more into detail regarding the geometry of hyperbolic
metric spaces. We prove the geometric mean value theorem for hyperbolic metric
spaces (Section 4.2), the existence of geodesic rays connecting two points in the
boundary of a CAT(-1) space (Proposition 4.4.4), and various geometrical theorems
regarding the sets
Shadz (x, σ) := {ξ ∈ ∂X : hx|ξiz ≤ σ},
which we call “shadows” due to their similarity to the famous shadows of Sullivan
[161, Fig. 2] on the boundary of Hd (Section 4.5). We remark that most proofs
of the existence of geodesics between points on the boundary of complete CAT(-1)
spaces, e.g. [39, Proposition II.9.32], assume properness and make use of it in a
crucial way, whereas we make no such assumption in Proposition 4.4.4. Finally,
in Section 4.6 we introduce “generalized polar coordinates” in a hyperbolic metric
space. These polar coordinates tell us that the action of a loxodromic isometry
(see Definition 6.1.2) on a hyperbolic metric space is roughly the same as the map
x 7→ λx in the upper half-plane E2 .

1.1. PRELIMINARIES

xxv

1.1.3. Discreteness. The first step towards extending the theory of Kleinian
groups to infinite dimensions (or more generally to hyperbolic metric spaces) is to
define the appropriate class of groups to consider. This is less trivial than might be
expected. Recalling that a d-dimensional Kleinian group is defined to be a discrete
subgroup of Isom(Hd ), we would want to define an infinite-dimensional Kleinian
group to be a discrete subgroup of Isom(H∞ ). But what does it mean for a subgroup
of Isom(H∞ ) to be discrete? In finite dimensions, the most natural definition is
to call a subgroup discrete if it is discrete relative to the natural topology on
Isom(Hd ); this definition works well since Isom(Hd ) is a Lie group. But in infinite
dimensions and especially in more exotic spaces, many applications require stronger
hypotheses (e.g., Theorem 1.2.1, Chapter 12). In Chapter 5, we discuss several
potential definitions of discreteness, which are inequivalent in general but agree in
the case of finite-dimensional space X = Hd (Proposition 5.2.10):
Definitions 5.2.1 and 5.2.6. Fix G ≤ Isom(X).
• G is called strongly discrete (SD) if for every bounded set B ⊆ X, we have
#{g ∈ G : g(B) ∩ B 6= } < ∞.
• G is called moderately discrete (MD) if for every x ∈ X, there exists an
open set U containing x such that
#{g ∈ G : g(U ) ∩ U 6= } < ∞.
• G is called weakly discrete (WD) if for every x ∈ X, there exists an open
set U containing x such that

g(U ) ∩ U 6=  ⇒ g(x) = x.
• G is called COT-discrete (COTD) if it is discrete as a subset of Isom(X)
when Isom(X) is given the compact-open topology (COT).
• If X is an algebraic hyperbolic space, then G is called UOT-discrete
(UOTD) if it is discrete as a subset of Isom(X) when Isom(X) is given
the uniform operator topology (UOT; cf. Section 5.1).
As our naming suggests, the condition of strong discreteness is stronger than
the condition of moderate discreteness, which is in turn stronger than the condition
of weak discreteness (Proposition 5.2.4). Moreover, any moderately discrete group
is COT-discrete, and any weakly discrete subgroup of Isom(H∞ ) is COT-discrete
(Proposition 5.2.7). These relations and more are summarized in Table 1 on p. 93.
Out of all these definitions, strong discreteness should perhaps be thought of as
the best generalization of discreteness to infinite dimensions. Thus, we propose that
the phrase “infinite-dimensional Kleinian group” should mean “strongly discrete

xxvi

1. INTRODUCTION AND OVERVIEW

subgroup of Isom(H∞ )”. However, in this monograph we will be interested in the
consequences of all the different notions of discreteness, as well as the interactions
between them.
Remark 1.1.6. Strongly discrete groups are known in the literature as metrically proper, and moderately discrete groups are known as wandering. However,
we prefer our terminology since it more clearly shows the relationship between the
different notions of discreteness.

1.1.4. The classification of semigroups. After clarifying the different types
of discreteness which can occur in infinite dimensions, we turn to the question of
classification. This question makes sense both for individual isometries and for entire semigroups.4 Historically, the study of classification began in the 1870s when
Klein proved a theorem classifying isometries of H2 and attached the words “elliptic”, “parabolic”, and “hyperbolic” to these classifications. Elliptic isometries are
those which have at least one fixed point in the interior, while parabolic isometries
have exactly one fixed point, which is a neutral fixed point on the boundary, and
hyperbolic isometries have two fixed points on the boundary, one of which is attracting and one of which is repelling. Later, the word “loxodromic” was used to
refer to isometries in H3 which have two fixed points on the boundary but which
are geometrically “screw motions” rather than simple translations. In what follows
we use the word “loxodromic” to refer to all isometries of Hn (or more generally a
hyperbolic metric space) with two fixed points on the boundary – this is analogous
to calling a circle an ellipse. Our real reason for using the word “loxodromic” in
this instance, rather than “hyperbolic”, is to avoid confusion with the many other
meanings of the word “hyperbolic” that have entered usage in various scenarios.
To extend this classification from individual isometries to groups, we call a
group “elliptic” if its orbits are bounded, “parabolic” if it has a unique neutral
global fixed point on the boundary, and “loxodromic” if it contains at least one
loxodromic isometry. The main theorem of Chapter 6 (viz. Theorem 6.2.3) is that
every subsemigroup of Isom(X) is either elliptic, parabolic, or loxodromic.
Classification of groups has appeared in the literature in various contexts, from
Eberlein and O’Neill’s results regarding visiblility manifolds [69], through Gromov’s remarks about groups acting on strictly convex spaces [83, §3.5] and wordhyperbolic groups [85, §3.1], to the more general results of Hamann [88, Theorem

4In Chapters 6-10, we work in the setting of semigroups rather than groups. Like dropping the

assumption of geodesicity (cf. Remark 1.1.4), this is done partly in order to broaden our class of
examples and partly for methodological reasons – we want to show exactly where the assumption
of being closed under inverses is being used. It should be also noted that semigroups sometimes
show up naturally when one is studying groups; cf. Proposition 10.5.4(B).

1.1. PRELIMINARIES

xxvii

2.7], Osin [140, §3], and Caprace, de Cornulier, Monod, and Tessera [48, §3.A]

regarding geodesic hyperbolic metric spaces.5 Many of these theorems have similar
statements to ours ([88] and [48] seem to be the closest), but we have not kept
track of this carefully, since our proof appears to be sufficiently different to warrant
independent interest anyway.
After proving Theorem 6.2.3, we discuss further aspects of the classification
of groups, such as the further classification of loxodromic groups given in §6.2.3:
a loxodromic group is called “lineal”, “focal”, or “of general type” according to
whether it has two, one, or zero global fixed points, respectively. (This terminology
was introduced in [48].) The “focal” case is especially interesting, as it represents a
class of nonelementary groups which have global fixed points.6 We show that certain
classes of discrete groups cannot be focal (Proposition 6.4.1), which explains why
such groups do not appear in the theory of Kleinian groups. On the other hand, we
show that in infinite dimensions, focal groups can have interesting limit sets even
though they satisfy only a weak form of discreteness; cf. Remark 13.4.3.
1.1.5. Limit sets. An important invariant of a Kleinian group G is its limit
set Λ = ΛG , the set of all accumulation points of the orbit of any point in the
interior. By putting an appropriate topology on the bordification of our hyperbolic
metric space X (§3.4.2), we can generalize this definition to an arbitrary subsemigroup of Isom(X). Many results generalize relatively straightforwardly7 to this new
context, such as the minimality of the limit set (Proposition 7.4.1) and the connection between classification and the cardinality of the limit set (Proposition 7.3.1).
In particular, we call a semigroup elementary if its limit set is finite.
In general, the convex hull of the limit set may need to be replaced by a
quasiconvex hull (cf. Definition 7.5.1), since in certain cases the convex hull does
not accurately reflect the geometry of the group. Indeed, Ancona [9, Corollary
C] and Borbely [32, Theorem 1] independently constructed examples of CAT(-1)
three-manifolds X for which there exists a point ξ ∈ ∂X such that the convex hull

of any neighborhood of ξ is equal to bord X. Although in a non-proper setting the

limit set may no longer be compact, compactness of the limit set is a reasonable
geometric condition that is satisfied for many examples of subgroups of Isom(H∞ )

5We remark that the results of [48, §3.A] can be generalized to non-geodesic hyperbolic metric

spaces by using the Bonk–Schramm embedding theorem [31, Theorem 4.1] (see also [29, Corollary
A.10]).
6Some sources (e.g. [148, §5.5]) define nonelementarity in a way such that global fixed points are
automatically ruled out, but this is not true of our definition (Definition 7.3.2).
7As is the case for many of our results, the classical proofs use compactness in a crucial way –
so here “straightforwardly” means that the statements of the theorems themselves do not require
modification.

xxviii

1. INTRODUCTION AND OVERVIEW

(e.g. Examples 13.2.2, 13.4.2). We call this condition compact type (Definition
7.7.1).
1.2. The Bishop–Jones theorem and its generalization
The term Poincaré series classically referred to a variety of averaging procedures, initiated by Poincaré in his aforementioned Acta memoirs, with a view
towards uniformization of Riemann surfaces via the construction of automorphic
b→C
b with no poles
forms. Given a Fuchsian group Γ and a rational function H : C

on ∂B2 , Poincaré proved that for every m ≥ 2 the series
X
H(γ(z))(γ ′ (z))m
γ∈Γ

(defined for z outside the limit set of Γ) converges uniformly to an automorphic
form of dimension m; see [63, p.218]. Poincaré called these series “θ-fuchsian series
of order m”, but the name “Poincaré series” was later used to refer to such objects.8
The question of for which m < 2 the Poincaré series still converges was investigated
by Schottky, Burnside, Fricke, and Ritter; cf. [2, pp.37-38].
In what would initially appear to be an unrelated development, mathematicians
began to study the “thickness” of the limit set of a Fuchsian group: in 1941 Myrberg
[135] showed that the limit set Λ of a nonelementary Fuchsian group has positive
logarithmic capacity; this was improved by Beardon [17] who showed that Λ has
positive Hausdorff dimension, thus deducing Myrberg’s result as a corollary (since
positive Hausdorff dimension implies positive logarithmic capacity for compact subsets of R2 [166]). The connection between this question and the Poincaré series was
first observed by Akaza, who showed that if G is a Schottky group for which the
Poincaré series converges in dimension s, then the Hausdorff s-dimensional measure
of Λ is zero [5, Corollary of Theorem A]. Beardon then extended Akaza’s result to
finitely generated Fuchsian groups [19, Theorem 5], as well as defining the exponent
of convergence (or Poincaré exponent ) δ = δG of a Fuchsian or Kleinian group to
be the infimum of s for which the Poincaré series converges in dimension s (cf. Definition 8.1.1 and [18]). The reverse direction was then proven by Patterson [142]
using a certain measure on Λ to produce the lower bound, which we will say more
about below in §1.4. Patterson’s results were then generalized by Sullivan [161] to

the setting of geometrically finite Kleinian groups. The necessity of the geometrical finiteness assumption was demonstrated by Patterson [143], who showed that
there exist Kleinian groups of the first kind (i.e. with limit set equal to ∂Hd ) with
8The modern definition of Poincaré series (cf. Definition 8.1.1) is phrased in terms of hyperbolic

geometry rather than complex analysis, but it agrees with the special case of Poincaré’s original
definition which occurs when H ≡ 1 and z = 0, with the caveat that γ ′ (z)m should be replaced
by |γ ′ (z)|m .

1.2. THE BISHOP–JONES THEOREM AND ITS GENERALIZATION

xxix

arbitrarily small Poincaré exponent [143] (see also [100] or [157, Example 8] for
an earlier example of the same phenomenon).
Generalizing these theorems beyond the geometrically finite case requires the
introduction of the radial and uniformly radial limit sets. In what follows, we will
denote these sets by Λr and Λur , respectively. Note that the radial and uniformly
radial limit sets as well as the Poincaré exponent can all (with some care) be defined
for general hyperbolic metric spaces; see Definitions 7.1.2, 7.2.1, and 8.1.1. The
radial limit set was introduced by Hedlund in 1936 in his analysis of transitivity of
horocycles [90, Theorem 2.4].
After some intermediate results [72, 158], Bishop and Jones [28, Theorem
1] generalized Patterson and Sullivan by proving that if G is a nonelementary
Kleinian group, then dimH (Λr ) = dimH (Λur ) = δ.9 Further generalization was
made by Paulin [144], who proved the equation dimH (Λr ) = δ in the case where
G ≤ Isom(X), and X is either a word-hyperbolic group, a CAT(-1) manifold, or a

locally finite unweighted simplicial tree which admits a discrete cocompact action.
We may now state the first major theorem of this monograph, which generalizes all
the aforementioned results:

that

Theorem 1.2.1. Let G ≤ Isom(X) be a nonelementary group. Suppose either
(1) G is strongly discrete,
(2) X is a CAT(-1) space and G is moderately discrete,
(3) X is an algebraic hyperbolic space and G is weakly discrete, or that
(4) X is an algebraic hyperbolic space and G acts irreducibly (cf. Section 7.6)
and is COT-discrete.

Then there exists σ > 0 such that
(1.2.1)

dimH (Λr ) = dimH (Λur ) = dimH (Λur ∩ Λr,σ ) = δ

(cf. Definitions 7.1.2 and 7.2.1 for the definition of Λr,σ ); moreover, for every
0 < s < δ there exist τ > 0 and an Ahlfors s-regular 10 set Js ⊆ Λur,τ ∩ Λr,σ .
For the proof of Theorem 1.2.1, see the comments below Theorem 1.2.3.
Remark. We note that weaker versions of Theorem 1.2.1 already appeared in
[58] and [73], each of which has a two-author intersection with the present paper.
In particular, case (1) of Theorem 1.2.1 appeared in [73] and the proofs of Theorem
1.2.1 and [73, Theorem 5.9] contain a number of redundancies. This was due to the
9Although Bishop and Jones’ theorem only states that dim (Λ ) = δ, they remark that their
r
H

proof actually shows that dimH (Λur ) = δ [28, p.4].

10Recall that a measure µ on a metric space Z is called Ahlfors s-regular if for all z ∈ Z and

0 < r ≤ 1, we have that µ(B(z, r)) ≍× r s . The topological support of an Ahlfors s-regular
measure is called an Ahlfors s-regular set.

xxx

1. INTRODUCTION AND OVERVIEW

fact that we worked on two projects which, despite having fundamentally different
objectives, both required essentially the same argument to produce “large, nice”
subsets of the limit set: in the present monograph, this argument forms the core of
the proof of our generalization of the Bishop–Jones theorem, while in [73], the main
use of the argument is in proving the full dimension of the set of badly approximable
points, in two different senses of the phrase “badly approximable” (approximation
by the orbits of distinguished points, vs. approximation by rational vectors in an
ambient Euclidean space). There are also similarities between the proof of Theorem
1.2.1 and the proof of the weaker version found in [58, Theorem 8.13], although
in this case the presentation is significantly different. However, we remark that
the main Bishop–Jones theorem of this monograph, Theorem 1.2.3, is significantly
more powerful than both [73, Theorem 5.9] and [58, Theorem 8.13].
Remark. The “moreover” clause is new even in the case which Bishop and
Jones considered, demonstrating that the limit set Λur can be approximated by
subsets which are particularly well distributed from a geometric point of view. It
does not follow from their theorem since a set could have large Hausdorff dimension
without having any closed Ahlfors regular subsets of positive dimension (much less
full dimension); in fact it follows from the work of Kleinbock and Weiss [116] that
the set of well approximable numbers forms such a set.11 In [73], a slight strengthening of this clause was used to deduce the full dimension of badly approximable
vectors in the radial limit set of a Kleinian group [73, Theorem 9.3].
Remark. It is possible for a group satisfying one of the hypotheses of Theorem
1.2.1 to also satisfy δ = ∞ (Examples 13.2.1-13.3.3 and 13.5.1-13.5.2);12 note that

Theorem 1.2.1 still holds in this case.

Remark. A natural question is whether (1.2.2) can be improved by showing
that there exists some σ > 0 for which dimH (Λur,σ ) = δ (cf. Definitions 7.1.2 and
7.2.1 for the definition of Λur,σ ). The answer is negative. For a counterexample,
take X = H2 and G = SL2 (Z) ≤ Isom(X); then for all σ > 0 there exists ε > 0

such that Λur,σ ⊆ BA(ε), where BA(ε) denotes the set of all real numbers with
Lagrange constant at most 1/ε. (This follows e.g. from making the correspondence
in [73, Observation 1.15 and Proposition 1.21] explicit.) It is well-known (see e.g.
[118] for a more precise result) that dimH (BA(ε)) < 1 for all ε > 0, demonstrating
that dimH (Λur,σ ) < 1 = δ.
11It could be objected that this set is not closed and therefore should not constitute a counterexample. However, since it has full measure, it has closed subsets of arbitrarily large measure (which
in particular still have dimension 1).
12For the parabolic examples, take a Schottky product (Definition 10.2.1) with a lineal group
(Definition 6.2.13) to get a nonelementary group, as suggested at the beginning of Chapter 13.

1.2. THE BISHOP–JONES THEOREM AND ITS GENERALIZATION

xxxi

Remark. Although Theorem 1.2.1 computes the Hausdorff dimension of the
radial and uniformly radial limit sets, there are many other subsets of the limit set
whose Hausdorff dimension it does not compute, such as the horospherical limit set
(cf. Definitions 7.1.3 and 7.2.1) and the “linear escape” sets (Λα )α∈(0,1) [122]. We
plan on discussing these issues at length in [57].
Finally, let us also remark that the hypotheses (1) - (4) cannot be weakened in
any of the obvious ways:
Proposition 1.2.2. We may have dimH (Λr ) < δ even if:
(1) G is moderately discrete (even properly discontinuous) (Example 13.4.4).
(2) X is a proper CAT(-1) space and G is weakly discrete (Example 13.4.1).
(3) X = H∞ and G is COT-discrete (Example 13.4.9).
(4) X = H∞ and G is irreducible and UOT-discrete (Example 13.4.2).
(5) X = H2 (Example 13.4.5).
In each case the counterexample group G is of general type (see Definition 6.2.13)
and in particular is nonelementary.
1.2.1. The modified Poincaré exponent. The examples of Proposition
1.2.2 illustrate that the Poincaré exponent does not always accurately calculate
the Hausdorff dimension of the radial and uniformly radial limit sets. In Chapter 8
we introduce a modified version of the Poincaré exponent which succeeds at accurately calculating dimH (Λr ) and dimH (Λur ) for all nonelementary groups G. (When
G is an elementary group, dimH (Λr ) = dimH (Λur ) = 0, so there is no need for a
sophisticated calculation in this case.) Some motivation for the following definition
is given in §8.2.
Definition 8.2.3. Let G be a subsemigroup of Isom(X).
• For each set S ⊆ X and s ≥ 0, let
X
Σs (S) =
b−skxk
x∈S

∆(S) = {s ≥ 0 : Σs (S) = ∞}
δ(S) = sup ∆(S).

• The modified Poincaré set of G is the set
\\
eG =
(8.2.2)
∆
∆(Sρ ),
ρ>0 Sρ

where the second intersection is taken over all maximal ρ-separated sets
Sρ ⊆ G(o).

xxxii

1. INTRODUCTION AND OVERVIEW

e G is called the modified Poincaré exponent of G.
• The number δeG = sup ∆
e G , we say that G is of generalized divergence type,13 while if
If δeG ∈ ∆
e G , we say that G is of generalized convergence type. Note
δeG ∈ [0, ∞) \ ∆

that if δeG = ∞, then G is neither of generalized convergence type nor of
generalized divergence type.

We may now state the most powerful version of our Bishop–Jones theorem:
Theorem 1.2.3 (Proven in Chapter 9). Let G  Isom(X) be a nonelementary
semigroup. There exists σ > 0 such that
(1.2.2)

e
dimH (Λr ) = dimH (Λur ) = dimH (Λur ∩ Λr,σ ) = δ.

Moreover, for every 0 < s < δe there exist τ > 0 and an Ahlfors s-regular set
Js ⊆ Λur,τ ∩ Λr,σ .

Theorem 1.2.1 can be deduced as a corollary of Theorem 1.2.3; specifically,
Propositions 8.2.4(ii) and 9.3.1 show that any group satisfying the hypotheses of
e and hence for such a group (1.2.2) implies (1.2.1).
Theorem 1.2.1 satisfies δ = δ,
On the other hand, Proposition 1.2.2 shows that Theorem 1.2.3 applies in many
cases where Theorem 1.2.1 does not.

We call a group Poincaré regular if its Poincaré exponent δ and modified
Poincaré exponent δe are equal. In this language, Proposition 9.3.1/Theorem 1.2.1

describes sufficient conditions for a group to be Poincaré regular, and Proposition
1.2.2 provides a list of examples of groups which are Poincaré irregular.
Though Theorem 1.2.3 requires G to be nonelementary, the following corollary
does not:
Corollary 1.2.4. Fix G  Isom(X). Then for some σ > 0,
(1.2.3)

dimH (Λr ) = dimH (Λur ) = dimH (Λur ∩ Λr,σ ).

Proof. If G is nonelementary, then (1.2.3) follows from (1.2.2). On the other
hand, if G is elementary, then all three terms of (1.2.3) are equal to zero.


1.3. Examples
A theory of groups acting on infinite-dimensional space would not be complete
without some good ways to construct examples. Techniques used in the finitedimensional setting, such as arithmetic construction of lattices and Dehn surgery,
do not work in infinite dimensions. (The impossibility of constructing lattices in
13We use the adjective “generalized” rather than “modified” because all groups of convergence/divergence type are also of generalized convergence/divergence type; see Corollary 8.2.8
below.

1.3. EXAMPLES

xxxiii

Isom(H∞ ) as a direct limit of arithmetic lattices in Isom(Hd ) is due to known lower
bounds on the covolumes of such lattices which blow up as the dimension goes
to infinity; see Proposition 12.2.3 below.) Nevertheless, there is a wide variety of
groups acting on H∞ , including many examples of actions which have no analogue
in finite dimensions.
1.3.1. Schottky products. The most basic tool for constructing groups or
semigroups on hyperbolic metric spaces is the theory of Schottky products. This
theory was created by Schottky in 1877 when he considered the Fuchsian group
generated by a finite collection of loxodromic isometries gi described by a disjoint
collection of balls Bi+ and Bi− with the property that gi (H2 \ Bi− ) = Bi+ . It was
extended further in 1883 by Klein’s Ping-Pong Lemma, and used effectively by
Patterson [143] to construct a “pathological” example of a Kleinian group of the
first kind with arbitrarily small Poincaré exponent.
We consider here a quite general formulation of Schottky products: a collection
of subsemigroups of Isom(X) is said to be in Schottky position if open sets can be
found satisfying the hypotheses of the Ping-Pong lemma whose closure is not equal
to X (cf. Definition 10.2.1). This condition is sufficient to guarantee that the
product of groups in Schottky position (called a Schottky product ) is always COTdiscrete, but stronger hypotheses are necessary in order to prove stronger forms of
discreteness. There is a tension here between hypotheses which are strong enough to
prove useful theorems and hypotheses which are weak enough to admit interesting
examples. For the purposes of this monograph we make a fairly strong assumption
(the strong separation condition, Definition 10.3.1), one which rules out infinitely
generated Schottky groups whose generating regions have an accumulation point
(for example, infinitely generated Schottky subgroups of Isom(Hd )). However, we
plan on considering weaker hypotheses in future work [57].
One theorem of significance in Chapter 10 is Theorem 10.4.7, which relates
the limit set of a Schottky product to the limit set of its factors together with the
image of a Cantor set ∂Γ under a certain symbolic coding map π : ∂Γ → ∂X.
As a consequence, we deduce that the properties of compact type and geometrical finiteness are both preserved under finite strongly separated Schottky products
(Corollary 10.4.8 and Proposition 12.4.19, respectively). A result analogous to Theorem 10.4.7 in the setting of infinite alphabet conformal iterated function systems
can be found in [128, Lemma 2.1].
In §10.5, we discuss some (relatively) explicit constructions of Schottky groups,

showing that Schottky products are fairly ubiquitous - for example, any two groups
which act properly discontinuously at some point of ∂X may be rearranged to be in
Schottky position, assuming that X is sufficiently symmetric (Proposition 10.5.1).

xxxiv

1. INTRODUCTION AND OVERVIEW

1.3.2. Parabolic groups. A major point of departure where the theory of
subgroups of Isom(H∞ ) becomes significantly different from the finite-dimensional
theory is in the study of parabolic groups. As a first example, knowing that a group
admits a discrete parabolic action on Isom(X) places strong restrictions on the algebraic properties of the group if X = HdF , but not if X = H∞
F . Concretely, discrete
parabolic subgroups of Isom(HdF ) are always virtually nilpotent (virtually abelian
if F = R), but any group with the Haagerup property admits a parabolic strongly
discrete action on H∞ (indeed, this is a reformulation of one of the equivalent definitions of the Haagerup property; cf. [50, p.1, (4)]). Examples of groups with
the Haagerup property include all amenable groups and free groups. Moreover,
strongly discrete parabolic subgroups of Isom(H∞ ) need not be finitely generated;
cf. Example 11.2.20.
Moving to infinite dimensions changes not only the algebraic but also the geometric properties of parabolic groups. For example, the cyclic group generated
by a parabolic isometry may fail to be discrete in any reasonable sense (Example
11.1.12), or it may be discrete in some senses but not others (Example 11.1.14).
The Poincaré exponent of a parabolic subgroup of Isom(HdF ) is always a half-integer
[54, Proof of Lemma 3.5], but the situation is much more complicated in infinite
dimensions. We prove a general lower bound on the Poincaré exponent of a parabolic subgroup of Isom(X) for any hyperbolic metric space X, depending only on
the algebraic structure of the group (Theorem 11.2.6); in particular, the Poincaré
exponent of a parabolic action of Zk on a hyperbolic metric space is always at least
k/2. Of course, it is well-known that all parabolic actions of Zk on Hd achieve
equality. By contrast, we show that for every δ > k/2 there exists a parabolic
action of Zk on H∞ whose Poincaré exponent is equal to δ (Theorem 11.2.11).

1.3.3. Geometrically finite and convex-cobounded groups. It has been
known for a long time that every finitely generated Fuchsian group has a finite-sided
convex fundamental domain (e.g. [108, Theorem 4.6.1]). This result does not generalize beyond two dimensions (e.g. [25, 102]), but subgroups of Isom(H3 ) with
finite-sided fundamental domains came to be known as geometrically finite groups.
Several equivalent definitions of geometrical finiteness in the three-dimensional setting became known, for example Beardon and Maskit’s condition that the limit set
is the union of the radial limit set Λr with the set Λbp of bounded parabolic points
[21], but the situation in higher dimensions was somewhat murky until Bowditch
[34] wrote a paper which described which equivalences remain true in higher dimensions, and which do not. The condition of a finite-sided convex fundamental
domain is no longer equivalent to any other conditions in higher dimensions (e.g.

1.3. EXAMPLES

xxxv

[12]), so a higher-dimensional Kleinian group is said to be geometrically finite if it
satisfies any of Bowditch’s five equivalent conditions (GF1)-(GF5).
In infinite dimensions, conditions (GF3)-(GF5) are no longer useful (cf. Remark
12.4.6), but appropriate generalizations of conditions (GF1) (convex core is equal
to a compact set minus a finite number of cusp regions) and (GF2) (the Beardon–
Maskit formula Λ = Λr ∪Λbp ) are still equivalent for groups of compact type. In fact,
(GF1) is equivalent to (GF2) + compact type (Theorem 12.4.5). We define a group
to be geometrically finite if it satisfies the appropriate analogue of (GF1) (Definition
12.4.1). A large class of examples of geometrically finite subgroups of Isom(H∞ )
is furnished by combining the techniques of Chapters 10 and 11; specifically, the
strongly separated Schottky product of any finite collection of parabolic groups
and/or cyclic loxodromic groups is geometrically finite (Corollary 12.4.20).
It remains to answer the question of what can be proven about geometrically
finite groups. This is a quite broad question, and in this monograph we content
ourselves with proving two theorems. The first theorem, Theorem 12.4.14, is a
generalization of the Milnor–Schwarz lemma [39, Proposition I.8.19] (see also Theorem 12.2.12), and describes both the algebra and geometry of a geometrically finite
group G: firstly, G is generated by a finite subset F ⊆ G together with a finite

collection of parabolic subgroups Gξ (which are not necessarily finitely generated,
e.g. Example 11.2.20), and secondly, the orbit map g 7→ g(o) is a quasi-isometric
embedding from (G, dG ) into X, where dG is a certain weighted Cayley metric (cf.
S
Example 3.1.2 and (12.4.6)) on G whose generating set is F ∪ ξ Gξ . As a conse-

quence (Corollary 12.4.17), we see that if the groups Gξ , ξ ∈ Λbp , are all finitely
generated, then G is finitely generated, and if these groups have finite Poincaré
exponent, then G has finite Poincaré exponent.
1.3.4. Counterexamples. A significant class of subgroups of Isom(H∞ ) that
has no finite-dimensional analogue is provided by the Burger–Iozzi–Monod (BIM)
representation theorem [40, Theorem 1.1], which states that any unweighted simplicial tree can be equivariantly and quasi-isometrically embedded into an infinitedimensional real hyperbolic space, with a precise relation between distances in the
domain and distances in the range. We call the embeddings provided by their
theorem BIM embeddings, and the corresponding homomorphisms provided by the
equivariance we call BIM representations. We generalize the BIM embedding theorem to the case where X is a separable R-tree rather than an unweighted simplicial
tree (Theorem 13.1.1).
If we have an example of an R-tree X and a subgroup Γ ≤ Isom(X) with a
certain property, then the image of Γ under a BIM representation generally has
the same property (Remark 13.1.4). Thus, the BIM embedding theorem allows

xxxvi

1. INTRODUCTION AND OVERVIEW

us to translate counterexamples in R-trees into counterexamples in H∞ . For example, if Γ is the free group on two elements acting on its Cayley graph, then
the image of Γ under a BIM representation provides a counterexample both to
an infinite-dimensional analogue of Margulis’s lemma (cf. Example 13.1.5) and to
an infinite-dimensional analogue of I. Kim’s theorem regarding length spectra of
finite-dimensional algebraic hyperbolic spaces (cf. Remark 13.1.6).
Most of the other examples in Chapter 13 are concerned with our various
notions of discreteness (cf. §1.1.3 above), the notion of Poincaré regularity (i.e.
e and the relations between them. Specifically, we show that
whether or not δ = δ),

the only relations are the relations which were proven in Chapter 5 and Proposition 9.3.1, as summarized in Table 1, p.93. Perhaps the most interesting of the
counterexamples we give is Example 13.4.2, which is the image under a BIM representation of (a countable dense subgroup of) the automorphism group Γ of the
4-regular unweighted simplicial tree. This example is notable because discreteness
properties are not preserved under taking the BIM representation: specifically, Γ
is weakly discrete but its image under the BIM representation is not. It is also
interesting to try to visualize this image geometrically (cf. Figure 13.4.1).
1.3.5. R-trees and their isometry groups. Motivated by the BIM representation theorem, we discuss some ways of constructing R-trees which admit
natural isometric actions. Our first method is the cone construction, in which one
starts with an ultrametric space (Z, D) and builds an R-tree X as a “cone” over
Z. This construction first appeared in a paper of F. Choucroun [52], although it is
similar to several other known cone constructions: [85, 1.8.A.(b)], [168], [31, §7].
R-trees constructed by the cone method tend to admit natural parabolic actions,
and in Theorem 14.1.5 we provide a necessary and sufficient condition for a function
to be the orbital counting function of some parabolic group acting on an R-tree.

Our second method is to staple R-trees together to form a new R-tree. We give
sufficient conditions on a graph (V, E), a collection of R-trees (Xv )v∈V , and a collection of sets A(v, w) ⊆ Xv and bijections ψv,w : A(v, w) → A(w, v) ((v, w) ∈ E)
such that stapling the trees (Xv )v∈V along the isometries (ψv,w )(v,w)∈E yields an
R-tree (Theorem 14.4.4). In §14.5, we give three examples of the stapling construction, including looking at the cone construction as a special case of the stapling
construction. The stapling construction is somewhat similar to a construction of
G. Levitt [120].

1.4. Patterson–Sullivan theory
The connection between the Poincaré exponent δ of a Kleinian group and the
geometry of its limit set is not limited to Hausdorff dimension considerations such

1.4. PATTERSON–SULLIVAN THEORY

xxxvii

as those in the Bishop–Jones theorem. As we mentioned before, Patterson and
Sullivan’s proofs of the equality dimH (Λ) = δ for geometrically finite groups rely on
the construction of a certain measure on Λ, the Patterson–Sullivan measure, whose
Hausdorff dimension is also equal to δ. In addition to connecting the Poincaré
exponent and Hausdorff dimension, the Patterson–Sullivan measure also relates to
the spectral theory of the Laplacian (e.g. [142, Theorem 3.1], [161, Proposition
28]) and the geodesic flow on the quotient manifold [103]. An important property
of Patterson–Sullivan measures is conformality. Given s > 0, a measure µ on ∂Bd
is said to be s-conformal with respect to a discrete group G ≤ Isom(Bd ) if
Z
|g ′ (ξ)|s dµ(ξ) ∀g ∈ G ∀A ⊆ ∂Bd .
(1.4.1)
µ(g(A)) =
A

The Patterson–Sullivan theorem on the existence of conformal measures may now
be stated as follows: For every Kleinian group G, there exists a δ-conformal measure
on Λ, where δ is the Poincaré exponent of G and Λ is the limit set of G.
When dealing with “coarse” spaces such as arbitrary hyperbolic metric spaces,
it is unreasonable to expect equality in (1.4.1). Thus, a measure µ on ∂X is said
to be s-quasiconformal with respect to a group G ≤ Isom(X) if
Z
g′ (ξ)s dµ(ξ) ∀g ∈ G ∀A ⊆ ∂X.
µ(g(A)) ≍×
A

Here g ′ (ξ) denotes the upper metric derivative of g at ξ; cf. §4.2.2. We remark that
if X is a CAT(-1) space and G is countable, then every quasiconformal measure is
coarsely asymptotic to a conformal measure (Proposition 15.2.1).
In Chapter 15, we describe the theory of conformal and quasiconformal meae
sures in hyperbolic metric spaces. The main theorem is the existence of δ-conformal
measures for groups of compact type (Theorem 15.4.6). An important special case

of this theorem has been proven by Coornaert [53, Théorème 5.4] (see also [41,
§1], [152, Lemme 2.1.1]): the case where X is proper and geodesic and G satisfies

δ < ∞. The main improvement from Coornaert’s theorem to ours is the ability
to construct quasiconformal measures for Poincaré irregular (δe < δ = ∞) groups;

this improvement requires an argument using the class of uniformly continuous
functions on bord X.
The big assumption of Theorem 15.4.6 is the assumption of compact type.

All proofs of the Patterson–Sullivan theorem seem to involve taking a weak-*
limit of a sequence of measures in X and then proving that the limit measure
is (quasi)conformal, but how can we take a weak-* limit if the limit set is not
compact? In fact, Theorem 15.4.6 becomes false if you remove the assumption of
compact type. In Proposition 16.6.1, we construct a group acting on an R-tree and

xxxviii

1. INTRODUCTION AND OVERVIEW

satisfying δ < ∞ which admits no δ-conformal measure on its limit set, and then

use the BIM embedding theorem (Theorem 13.1.1) to get an example in H∞ .
Surprisingly, it turns out that if we replace the hypothesis of compact type with
the hypothesis of divergence type, then the theorem becomes true again. Specifically,
we have the following:
Theorem 1.4.1 (Proven in Chapter 16). Let G ≤ Isom(X) be a nonelementary
e
group of generalized divergence type (see Definition 8.2.3). Then there exists a δquasiconformal measure µ for G supported on Λ, where δe is the modified Poincaré
exponent of G. It is unique up to a multiplicative constant in the sense that if
µ1 , µ2 are two such measures then µ1 ≍× µ2 (cf. Remark 15.1.2). In addition, µ
is ergodic and gives full measure to the radial limit set of G.

To motivate Theorem 1.4.1, we recall the connection between the divergence
type condition and Patterson–Sullivan theory in finite dimensions. Although the
Patterson–Sullivan theorem guarantees the existence of a δ-conformal measure, it
does not guarantee its uniqueness. Indeed, the δ-conformal measure is often not
unique; see e.g. [10]. However, it turns out that the hypothesis of divergence type
is enough to guarantee uniqueness. In fact, the condition of divergence type turns
out to be quite important in the theory of conformal measures:
Theorem 1.4.2 (Hopf–Tsuji–Sullivan theorem, [138, Theorem 8.3.5]). Fix d ≥
2, let G ≤ Isom(Hd ) be a discrete group, and let δ be the Poincaré exponent of G.

Then for any δ-conformal measure µ ∈ M(Λ), the following are equivalent:
(A) G is of divergence type.
(B) µ gives full measure to the radial limit set Λr (G).
(C) G acts ergodically on (Λ, µ) × (Λ, µ).

In particular, if G is of divergence type, then every δ-conformal measure is ergodic,
so there is exactly one (ergodic) δ-conformal probability measure.
We remark that our sentence “In particular . . . ” stated in theorem above was
not included in [138, Theorem 8.3.5] but it is well-known and follows easily from
the equivalence of (A) and (C).
Remark 1.4.3. Theorem 1.4.2 has a long history. The equivalence (B) ⇔ (C)

was first proven by E. Hopf in the case δ = d − 114 [99, 100] (1936, 1939). The

equivalence (A) ⇔ (B) was proven by Z. Yûjôbô in the case δ = d − 1 = 1 [176]
(1949), following an incorrect proof by M. Tsuji [169] (1944).15 Sullivan proved (A)
⇔ (C) in the case δ = d − 1 [163, Theorem II], then generalized this equivalence
14In this paragraph, when we say that someone proves the case δ = d − 1, we mean that they

considered the case where µ is Hausdorff (d − 1)-dimensional measure on S d−1 .
15See [163, p.484] for some further historical remarks on the case δ = d − 1 = 1.

1.4. PATTERSON–SULLIVAN THEORY

xxxix

to the case δ > (d − 1)/2 [161, Theorem 32]. He also proved (B) ⇔ (C) in full

generality [161, Theorem 21]. Next, W. P. Thurston gave a simpler proof of (A)
⇒ (B)16 in the case δ = d − 1 [4, Theorem 4 of Section VII]. P. J. Nicholls finished

the proof by showing (A) ⇔ (B) in full generality [138, Theorems 8.2.2 and 8.2.3].
Later S. Hong re-proved (A) ⇒ (B) in full generality twice in two independent
papers [97, 98], apparently unaware of any previous results. Another proof of (A)
⇒ (B) in full generality, which was conceptually similar to Thurston’s proof, was
given by P. Tukia [171, Theorem 3A]. Further generalization was made by C. Yue
[175] to negatively curved manifolds, and by T. Roblin [151, Théorème 1.7] to
proper CAT(-1) spaces.
Having stated the Hopf–Tsuji–Sullivan theorem, we can now describe why Theorem 1.4.1 is true, first on an intuitive level and then giving a sketch of the real
proof. On an intuitive level, the fact that divergence type implies both “existence
and uniqueness” of the δ-conformal measure in finite dimensions indicates that perhaps the compactness assumption is not needed – the sequence of measures used
to construct the Patterson–Sullivan measure converges already, so it should not be
necessary to use compactness to take a convergent subsequence.
The real proof involves taking the Samuel–Smirnov compactification of bord X,
considered as a metric space with respect to a visual metric (cf. §3.6.3). The
Samuel–Smirnov compactification of a metric space (cf. [136, §7]) is conceptually
similar to the more familiar Stone–Čech compactification, except that only uniformly continuous functions on the metric space extend to continuous functions

on the compactification, not all continuous functions. If we used the Stone–Čech
compactification rather than the Samuel–Smirnov compactification, then our proof
would only apply to groups with finite Poincaré exponent; cf. Remark 16.1.3 and
Remark 16.3.5.
Sketch of the proof of Theorem 1.4.1. We denote the Samuel–Smirnov
b By a nonstandard analogue of Theorem 15.4.6
compactification of bord X by X.
e
d By a
(viz. Lemma 16.3.4), there exists a δ-quasiconformal
measure µ
b on ∂X.

generalization of Theorem 1.4.2 (viz. Proposition 16.4.1), µ
b gives full measure to
cr . But a simple computation (Lemma 16.2.5) shows that
the radial limit set Λ
c
Λr = Λr , demonstrating that µ
b ∈ M(Λ).


1.4.1. Quasiconformal measures of geometrically finite groups. Let
us consider a geometrically finite group G ≤ Isom(X) with Poincaré exponent
δ < ∞, and let µ be a δ-quasiconformal measure on Λ. Such a measure exists
since geometrically finite groups are of compact type (Theorem 12.4.5 and Theorem

16By this point, it was considered obvious that (B) ⇒ (A).

xl

1. INTRODUCTION AND OVERVIEW

15.4.6), and is unique as long as G is of divergence type (Corollary 16.4.6). When
X = Hd , the geometry of µ is described by the Global Measure Formula [165,
Theorem on p.271], [160, Theorem 2]: the measure of a ball B(η, e−t ) is coarsely
asymptotic to e−δt times a factor depending on the location of the point ηt := [o, η]t
in the quotient manifold Hd /G. Here [o, η]t is the unique point on the geodesic
connecting o and η with distance t from o; cf. Notations 3.1.6, 4.4.3.
In a general hyperbolic metric space X (indeed, already for X = H∞ ), one
cannot get a precise asymptotic for µ(B(η, e−t )), due to the fact that the measure
µ may fail to be doubling (Example 17.4.12). Instead, our version of the global
measure formula gives both an upper bound and a lower bound for µ(B(η, e−t )).
Specifically, we define a function m : Λ × [0, ∞) → (0, ∞) (for details see (17.2.1))
and then show:

Theorem 1.4.4 (Global measure formula, Theorem 17.2.2; proven in Section
17.3). For all η ∈ Λ and t > 0,
(1.4.2)

m(η, t + σ) .× µ(B(η, e−t )) .× m(η, t − σ),

where σ > 0 is independent of η and t.
It is natural to ask for which groups (1.4.2) can be improved to an exact asymptotic, i.e. for which groups µ is doubling. We address this question in Section
17.4, proving a general result (Proposition 17.4.8), a special case of which is that if
X is a finite-dimensional algebraic hyperbolic space, then µ is doubling (Example
17.4.11). Nevertheless, there are large classes of examples of groups G ≤ Isom(H∞ )

for which µ is not doubling (Example 17.4.12), illustrating once more the wide
difference between H∞ and its finite-dimensional counterparts.

It is also natural to ask about the implications of the Global Measure Formula
for the dimension theory of the measure µ. For example, when X = Hd , the Global
Measure Formula was used to show that dimH (µ) = δ [160, Proposition 4.10]. In
our case we have:
Theorem 1.4.5 (Cf. Theorem 17.5.9). If for all p ∈ P , the series
X
e−δkhk khk
(1.4.3)
h∈Gp

converges, then µ is exact dimensional (cf. Definition 17.5.2) of dimension δ. In
particular,
dimH (µ) = dimP (µ) = δ .
The hypothesis that (1.4.3) converges is a very non-restrictive hypothesis. For
example, it is satisfied whenever δ > δp for all p ∈ P (Corollary 17.5.10). Combining

1.5. APPENDICES

xli

with Proposition 10.3.10 shows that any counterexample must satisfy
X
X
e−δkhk khk
e−δkhk < ∞ =
h∈Gp

h∈Gp

for some p ∈ P , creating a very narrow window for the orbital counting function

Np (cf. Notation 17.2.1) to lie in. Nevertheless, we show that there exist counterexamples (Example 17.5.14) for which the series (1.4.3) diverges. After making
some simplifying assumptions, we are able to prove (Theorem 17.5.13) that the
Patterson–Sullivan measures of groups for which (1.4.3) diverges cannot be exact
dimensional, and in fact satisfy dimH (µ) = 0.
There is a relation between exact dimensionality of the Patterson–Sullivan measure and the theory of Diophantine approximation on the boundary of ∂X, as described in [73]. Specifically, if VWAξ denotes the set of points which are very well
approximable with respect to a distinguished point ξ (cf. §17.5.1), then we have
the following:
Theorem 1.4.6 (Cf. Theorem 17.5.8). The following are equivalent:
(A) For all p ∈ P , µ(VWAp ) = 0.
(B) µ is exact dimensional.
(C) dimH (µ) = δ.

(D) For all ξ ∈ Λ, µ(VWAξ ) = 0.
In particular, combining with Theorem 1.4.5 demonstrates that the equation
µ(VWAξ ) = 0
holds for a large class of geometrically finite groups G and for all ξ ∈ Λ. This
improves the results of [73, §1.5.3].
1.5. Appendices
We conclude this monograph with two appendices. Appendix A contains a list
of open problems, and Appendix B an index of defined terms.

Part 1

Preliminaries

This part will be divided as follows: In Chapter 2 we define the class of algebraic
hyperbolic spaces, which are often called rank one symmetric spaces of noncompact
type. In Chapters 3 and 4, we define the class of hyperbolic metric spaces and study
their geometry. In Chapter 5, we explore different notions of discreteness for groups
of isometries of a metric space. In Chapter 6 we prove two classification theorems,
one for isometries (Theorem 6.1.4) and one for semigroups of isometries (Theorem
6.2.3). Finally, in Chapter 7 we define and study the limit set of a semigroup of
isometries.

CHAPTER 2

Algebraic hyperbolic spaces
In this chapter we introduce our main objects of interest, algebraic hyperbolic spaces in finite and infinite dimensions. References for the theory of finitedimensional algebraic hyperbolic spaces, which are often called rank one symmetric
spaces of noncompact type, include [39, 45, 123]. Infinite-dimensional algebraic
hyperbolic spaces, as well as some non-hyperbolic infinite-dimensional symmetric
spaces, have been discussed in [67].
2.1. The definition
Finite-dimensional rank one symmetric spaces of noncompact type come in
four flavors, corresponding to the classical division algebras R, C, Q (quaternions),
and O (octonions).1 The first three division algebras have corresponding rank one
symmetric spaces of noncompact type of arbitrary dimension, but there is only one
rank one symmetric space of noncompact type corresponding to the octonions; it
occurs in dimension two (which corresponds to real dimension 16). Consequently,
the octonion rank one symmetric space of noncompact type (known as the Cayley
hyperbolic plane 2) does not have an infinite-dimensional analogue, while the other
three classes do admit infinite-dimensional analogues.
The rank one symmetric spaces of noncompact type corresponding to R have
constant negative curvature. However, those corresponding to the other division
algebras have variable negative curvature [147, Lemmas 2.3, 2.7, 2.11] (see also
[93, Corollary of Proposition 4]).
Remark 2.1.1. In this monograph we will use the term “algebraic hyperbolic
spaces” to refer to all rank one symmetric spaces of noncompact type except the
Cayley hyperbolic plane H2O , in order to avoid dealing with the complicated algebra of the octonions.3 However, we feel confident that all the theorems regarding
1We denote the quaternions by Q in order to avoid confusion with the rank one symmetric space

of noncompact type (defined over Q) itself, which we will denote by H. Be aware that Q should
not be confused with the set of rational numbers.
2Not to be confused with the Cayley plane, a different mathematical object.
3The complications come from the fact that the octonions are not associative, thus making it
somewhat unclear what it means to say that O3 is a vector space “over” the octonions, since in
general (xa)b 6= x(ab).
3

4

2. ALGEBRAIC HYPERBOLIC SPACES

algebraic hyperbolic spaces in this monograph can be generalized to the Cayley
hyperbolic plane (possibly after modifying the statements slightly). We leave this
task to an algebraist.
For the reader interested in learning more about the Cayley hyperbolic plane,
see [133, pp.136-139], [156], or [7]; see also [14] for an excellent introduction to
octonions in general.
Fix F ∈ {R, C, Q} and an index set J, and let us construct an algebraic hyper-

bolic space (i.e. a rank one symmetric space of noncompact type) over the field F
in dimension #(J). We remark that usually we will let J = N = {1, 2, . . .}, but

occasionally J may be an uncountable set. Let
(
H=

HFJ

:=

x = (xi )i∈J ∈ F

and for x ∈ H let
kxk :=

X
i∈J

X

J

|xi |2

i∈J

!1/2

2

|xi | < ∞

)

,

.

We will think of H as a right F-module, so scalars will always act on the right.4
Note that

kxak = |a| · kxk ∀x ∈ H ∀a ∈ F.
A sesquilinear form on H is an R-bilinear map B(·, ·) : H × H → F satisfying
B(xa, y) = aB(x, y) and B(x, ya) = B(x, y)a.5
Here and from now on a denotes the conjugate of a complex or quaternionic number
a ∈ F; if F = R, then a = a.
A sesquilinear form is said to be skew-symmetric if B(y, x) = B(x, y). For
example, the map
BE (x, y) :=

X

xi yi

i∈J

is a skew-symmetric sesquilinear form. Note that
E(x) := BE (x, x) = kxk2 .
2.2. The hyperboloid model
Assume that 0 ∈
/ J, and let


J∪{0}
J∪{0}
L = LF
= HF
= x = (xi )i∈J∪{0} ∈ FJ∪{0}


X

i∈J∪{0}

|xi |2 < ∞





.

4The advantage of this convention is that it allows operators to act on the left.
5In the case F = C, this disagrees with the usual convention; we follow here the convention of

[123, §3.3.1].

2.2. THE HYPERBOLOID MODEL

5

Consider the skew-symmetric sesquilinear form BQ : L × L → F defined by
X
xi yi
BQ (x, y) := −x0 y0 +
i∈J

and its associated quadratic form
(2.2.1)

Q(x) := BQ (x, x) = −|x0 |2 +

X
i∈J

|xi |2 .

We observe that the form Q is not positive definite, since Q(e0 ) = −1.
Remark 2.2.1. If F = R, then the form Q is called a Lorentzian quadratic

form, and the pair (L, Q) is called a Minkowski space.

Let P(L) denote the projectivization of L, i.e. the quotient of L \ {0} under

the equivalence relation x ∼ xa (x ∈ L \ {0}, a ∈ F \ {0}). Let
J∪{0}

H = HJF := {[x] ∈ P(LF

) : Q(x) < 0},

and consider the map dH : H × H → [0, ∞) defined by the equation
(2.2.2)

|BQ (x, y)|
cosh dH ([x], [y]) = p
, [x], [y] ∈ H.
|Q(x)| · |Q(y)|

Note that the map dH is well-defined because the right hand side is invariant under
multiplying x and y by scalars.
Proposition 2.2.2. The map dH is a metric on H that is compatible with the
natural topology, when viewed as a subspace of the quotient space P(L). Moreover,
for any two distinct points [x], [y] ∈ H there exists a unique isometric embedding
γ : R → H such that γ(0) = [x] and γ ◦ dH ([x], [y]) = [y].

Remark 2.2.3. The second sentence is the unique geodesic extension property
of H. It holds more generally for Riemannian manifolds (cf. Remark 2.2.7 below),
but is an important distinguishing feature in the larger class of uniquely geodesic
metric spaces.
Proof of Proposition 2.2.2. The key to the proof is the following lemma,
which may also be deduced from the infinite-dimensional analogue of Sylvester’s
law of inertia [124, Lemma 3].
Lemma 2.2.4. Fix z ∈ L with Q(z) < 0, and let z⊥ = {w : BQ (z, w) = 0}.
Then Q ↿ z⊥ is positive definite.
⊥

Proof of Lemma 2.2.4. By contradiction, suppose Q(y) ≤ 0 for some y ∈

z . There exist a, b ∈ F, not both zero, such that y0 a + z0 b = 0. But then
0 < Q(ya + zb) = |a|2 Q(y) + |b|2 Q(z) ≤ 0,
which provides a contradiction.

⊳

6

2. ALGEBRAIC HYPERBOLIC SPACES

Now fix [x], [y], [z] ∈ H, and let x, y, z ∈ L\{0} be representatives which satisfy
BQ (x, z) = BQ (y, z) = Q(z) = −1.
Then
1
cosh dH ([x], [z]) = p
1 − Q(x − z)
p
Q(x − z)
sinh dH ([x], [z]) = p
1 − Q(x − z)

1
cosh dH ([y], [z]) = p
1 − Q(y − z)
p
Q(y − z)
sinh dH ([y], [z]) = p
·
1 − Q(y − z)

By the addition law for hyperbolic cosine we have
p
p
1 + Q(x − z) Q(y − z)
p
·
cosh(dH ([x], [z]) + dH ([y], [z])) = p
1 − Q(x − z) 1 − Q(y − z)
On the other hand, we have

1
1
p
|−1 + BQ (x − z, y − z)| .
cosh dH ([x], [y]) = p
1 − Q(x − z) 1 − Q(y − z)

Since x− z, y − z ∈ z⊥ , the Cauchy–Schwartz inequality together with Lemma 2.2.4
gives

|−1 + BQ (x − z, y − z)| ≤ 1 +

p
p
Q(x − z) Q(y − z),

with equality if and only if x − z and y − z are proportional with a negative real
constant of proportionality. This demonstrates the triangle inequality.
To show that dH is compatible with the natural topology, it suffices to show

that if U is a neighborhood in the natural topology of a point [x] ∈ H, then there
exists ε > 0 such that B([x], ε) ⊆ U . Indeed, fix a representative x ∈ [x]; then there

exists δ > 0 such that ky − xk ≤ δ implies [y] ∈ U . Now, given [y] ∈ B([x], ε),
choose a representative y ∈ [y] such that z := y − x satisfies BQ (x, z) = 0; this

is possible since any representative y ∈ [y] satisfies BQ (x, y) 6= 0 by Lemma 2.2.4.
Then
s
Q(x)
|Q(x)|
·
cosh dH ([x], [y]) = p
=
Q(x) + Q(z)
|Q(x)| · |Q(x) + Q(z)|
So if dH ([x], [y]) ≤ ε, then Q(z) ≤ Q(x)[1 − 1/ cosh(ε)]. By Lemma 2.2.4, there
exists C > 0 such that kzk2 ≤ CQ(x)[1−1/ cosh(ε)]. In particular, we may choose ε

so that CQ(x)[1 − 1/ cosh(ε)] ≤ δ, which completes the proof that dH is compatible
with the natural topology.
Now suppose that γ : R → H is an isometric embedding, and let [z] = γ(0).
Choose a representative z ∈ L \ {0} such that Q(z) = −1, and for each t ∈ R \ {0}

choose a representative xt ∈ L \ {0} such that BQ (xt , z) = −1. The preceding
argument shows that for t1 < 0 < t2 , xt1 − z and xt2 − z are proportional with a

2.2. THE HYPERBOLOID MODEL

7

negative constant of proportionality. Together with (2.2.2), this implies that
(2.2.3)

xt = z + tanh(t)w

for some w ∈ z⊥ with Q(w) = 1. Conversely, direct calculation shows that the

equation (2.2.3) defines an isometric embedding γz,w : R → H via the formula
γz,w (t) = [xt ].

Definition 2.2.5. An algebraic hyperbolic space is a pair (HJF , dH ), where F ∈

{R, C, Q} and J is a nonempty set such that 0 ∈
/ J.

Remark 2.2.6. In finite dimensions, the class of algebraic hyperbolic spaces is
identical (modulo the Cayley hyperbolic plane, cf. Remark 2.1.1) to the class of
rank one symmetric spaces of noncompact type. This follows from the classification
theorem for finite-dimensional symmetric spaces, see e.g. [94, p.518].6 It is not clear
whether an analogous theorem holds in infinite dimensions (but see [68] for some
results in this direction).
Remark 2.2.7. In finite dimensions, the metric dH may be defined as the length
metric associated to a certain Riemannian metric on H; cf. [147, §2.2]. The same

procedure works in infinite dimensions; cf. [119] for an exposition of the theory
of infinite dimensional manifolds. Although a detailed account of the theory of
infinite-dimensional Riemannian manifolds would be too much of a digression, let
us make the following points:
• An infinite-dimensional analogue of the Hopf–Rinow theorem is false [13],
i.e. there exists an infinite-dimensional Riemannian manifold such that
some two points on that manifold cannot be connected by a geodesic.
However, if an infinite-dimensional Riemannian manifold X is nonpositively curved, then any two points of X can be connected by a unique
geodesic as a result of the infinite-dimensional Cartan–Hadamard theorem [119, IX, Theorem 3.8]; moreover, this geodesic is length-minimizing.
In particular, if one takes a Riemannian manifolds approach to defining
infinite-dimensional algebraic hyperbolic spaces, then the second assertion
of Proposition 2.2.2 follows from the Cartan–Hadamard theorem.
• A bijection between two infinite-dimensional Riemannian manifolds is an
isometry with respect to the length metric if and only if it is a diffeomorphism which induces an isometry on the Riemannian metric [76, Theorem
7]. This theorem is commonly known as the Myers–Steenrod theorem, as
S. B. Myers and N. E. Steenrod proved its finite-dimensional version [134].
The difficult part of this theorem is proving that any bijection which is an
6In the notation of [94], the spaces Hp , Hp , Hp , and H2 are written as SO(p, 1)/ SO(p),
O
Q
C
R

SU(p, 1)/ SU(p), Sp(p, 1)/ Sp(p), and (f4(−20) , so(9)), respectively.

8

2. ALGEBRAIC HYPERBOLIC SPACES

isometry with respect to the length metric is differentiable. In the case of
algebraic hyperbolic spaces, however, this follows directly from Theorem
2.3.3 below.

2.3. Isometries of algebraic hyperbolic spaces
We define the group of isometries of a metric space (X, d) to be the group
Isom(X) := {g : X → X : g is a bijection and d(g(x), g(y)) = d(x, y) ∀x, y ∈ X}.
In this section we will compute the group of isometries of an arbitrary algebraic
hyperbolic space. Fix F ∈ {R, C, Q} and an index set J, and let H = HFJ , L =
J∪{0}

LF

, and H = HJF . We begin with the following observation:

Observation 2.3.1. Let OF (L; Q) denote the group of Q-preserving F-linear

automorphisms of L. Then for all T ∈ OF (L; Q), the map [T ] : H → H defined by
the equation
(2.3.1)

[T ]([x]) = [T x]

is an isometry of (H, dH ).
Proof. The map [T ] is well-defined by the associativity property T (xa) =
(T x)a. Since T is Q-preserving and F-linear, the polarization identity (the three

versions cover the three cases when the base field F = R, C, and Q respectively)

1


 4 [Q(x + y) − Q(x − y)]


1
+ y) − Q(x − y) − iQ(x + yi) + iQ(x − yi)]
BQ (x, y) = 4 [Q(x
#
"



P

1
 Q(x + y) − Q(x − y) +
− ℓQ(x + yℓ) + ℓQ(x − yℓ)

4
ℓ=i,j,k

guarantees that
(2.3.2)

BQ (T x, T y) = BQ (x, y) ∀x, y ∈ H.

Comparing with (2.2.2) shows that [T ] is an isometry.



The group OF (L; Q) is quite large. In addition to containing all maps of the
form T ⊕ I, where T ∈ OF (H; E) and I : F → F is the identity map, it also contains

the so-called Lorentz boosts



xi


(2.3.3)
Tj,t (x) =  cosh(t)xj + sinh(t)x0



sinh(t)x + cosh(t)x
j
0

i 6= 0, j
i=j
i=0







i∈J∪{0}

, j ∈ J, t ∈ R.

2.3. ISOMETRIES OF ALGEBRAIC HYPERBOLIC SPACES

9

We leave it as an exercise that OF (H; E) ⊕ {I} and the Lorentz boosts in fact

generate the group OF (L; Q).

Observation 2.3.2. The group
POF (L; Q) = {[T ] : T ∈ OF (L; Q)} ≤ Isom(H)
acts transitively on H.
Proof. Let o = [(1, 0)]. The orbit of o under POF (L; Q) contains its image
under the Lorentz boosts. Specifically, for every t ∈ R the orbit of o contains the

point [(cosh(t), sinh(t), 0)]. Applying maps of the form [T ⊕I], T ∈ OF (H, E), shows
that the orbit of o is H.

We may ask the question of whether the group POF (L; Q) is equal to Isom(H)
or is merely a subgroup. The answer turns out to depend on the division algebra
F:

Theorem 2.3.3. If F ∈ {R, Q} then Isom(H) = POF (L; Q). If F = C, then
POF (L; Q) is of index 2 in Isom(H).
Remark 2.3.4. In finite dimensions, Theorem 2.3.3 is given as an exercise
in [39, Exercise II.10.21]. Because of the importance of Theorem 2.3.3 to this
monograph, we provide a full proof.
Before proving Theorem 2.3.3, it will be convenient for us to introduce a group
somewhat larger than OF (L; Q). Let Aut(F) denote the group of automorphisms

of F as an R-algebra, i.e.
(
Aut(F) =

σ:F→F

σ is an R-linear bijection and
σ(ab) = σ(a)σ(b) for all a, b ∈ F

)

.

We will say that an R-linear map T : L → L is F-skew linear if there exists
σ ∈ Aut(F) such that
(2.3.4)

T (xa) = T (x)σ(a) for all x ∈ H and a ∈ F.

The group of skew-linear bijections T : L → L which preserve Q will be denoted
O∗F (L; Q). For each T , the unique σ ∈ Aut(F) satisfying (2.3.4) will be denoted σT .
Note that the map T 7→ σT is a homomorphism.
Warning. The associative law (T x)a = T (xa) is not valid for T ∈ O∗F (L; Q);

rather, T (xa) = (T x)σT (a) by (2.3.4). Thus when discussing elements of O∗F (L; Q),

we must be careful of parentheses.

Example 2.3.5. For each σ ∈ Aut(F), the map
σ J (x) = (σ(xi ))i∈J

10

2. ALGEBRAIC HYPERBOLIC SPACES

is F-skew-linear and Q-preserving, and σσJ = σ.
Observation 2.3.6. For T ∈ O∗F (L; Q),
BQ (T x, T y) = σT (BQ (x, y)) ∀x, y ∈ L.
Proof. By (2.3.2), the formula holds when T ∈ OF (L; Q), and direct calcula-

tion shows that it holds when T = σ J for some σ ∈ Aut(F). Since O∗F (L; Q) is a
semidirect product of the groups OF (L; Q) and {σ J : σ ∈ Aut(F)}, this completes
the proof.



We observe that if T ∈ O∗F (L; Q), then (2.3.4) shows that T preserves F-lines,

i.e. T (xF) = T (x)F for all x ∈ L \ {0}. Thus the equation (2.3.1) defines a map
[T ] : H → H, which is an isometry by Observation 2.3.6. Thus if
PO∗F (L; Q) = {[T ] : T ∈ O∗F (L; Q)},
then
POF (L; Q) ≤ PO∗F (L; Q) ≤ Isom(H).
We are now ready to begin the

Proof of Theorem 2.3.3. The proof will consist of two parts. In the first,
we show that PO∗F (L; Q) = Isom(H), and in the second we show that POF (L; Q) is

equal to PO∗F (L; Q) if F = R, Q and is of index 2 in PO∗F (L; Q) if F = C.
Fix g ∈ Isom(H); we claim that g ∈ PO∗F (L; Q). Let z = (1, 0), and let o = [z].

By Observation 2.3.2, there exists [T ] ∈ POF (L; Q) such that [T ](o) = g(o). Thus,
we may without loss of generality assume that g(o) = o.
We observe that z⊥ = H. Let S(H) denote the unit sphere of H, i.e. S(H) =
{w ∈ H : Q(w) = 1}. For each w ∈ S(H), the embedding γz,w : R → H defined in

the proof of Proposition 2.2.2 is an isometry. By Proposition 2.2.2, its image under
g must also be an isometry. Specifically, there exists f (w) ∈ S(H) such that
(2.3.5)

g([z + tanh(t)w]) = [z + tanh(t)f (w)] ∀t ∈ R.

The fact that g is a bijection implies that f : S(H) → S(H) is a bijection. Moreover,

the fact that g is an isometry means that for all w1 , w2 ∈ S(H) and t1 , t2 ∈ R, we
have

d([z + tanh(t1 )w1 ], [z + tanh(t2 )w2 ]) = d([z + tanh(t1 )f (w1 )], [z + tanh(t2 )f (w2 )]).

2.3. ISOMETRIES OF ALGEBRAIC HYPERBOLIC SPACES

11

Recalling that
cosh d([z + tanh(t1 )w1 ], [z + tanh(t2 )w2 ])
|BQ (z + tanh(t1 )w1 , z + tanh(t2 )w2 )|
= p
|Q(z + tanh(t1 )w1 )| · |Q(z + tanh(t2 )w2 )|

=
we see that

| − 1 + tanh(t1 ) tanh(t2 )BQ (w1 , w2 )|
q

 ,
1 − tanh2 (t1 ) 1 − tanh2 (t2 )

| − 1 + tanh(t1 ) tanh(t2 )BQ (w1 , w2 )| = | − 1 + tanh(t1 ) tanh(t2 )BQ (f (w1 ), f (w2 ))|.
Write θ = tanh(t1 ) tanh(t2 ). Squaring both sides gives
LHS2
(2.3.6)

=

θ2 |BQ (w1 , w2 )|2 − 2θ Re[BQ (w1 , w2 )] + 1

=

θ2 |BQ (f (w1 ), f (w2 ))|2 − 2θ Re[BQ (f (w1 ), f (w2 ))] + 1.

||

RHS2

We observe that for w1 , w2 ∈ S(H) fixed, (2.3.6) holds for all −1 < θ < 1. In
particular, taking the first and second derivatives and plugging in θ = 0 gives
(2.3.7)

Re[BQ (w1 , w2 )] = Re[BQ (f (w1 ), f (w2 ))]

(2.3.8)

|BQ (w1 , w2 )| = |BQ (f (w1 ), f (w2 ))|.

Extend f to a bijection f : H → H by letting f (0) = 0 and f (tw) = tf (w) for
t > 0, w ∈ S(H). We observe that (2.3.7) and (2.3.8) hold also for the extended

version of f .

Claim 2.3.7. f is R-linear.
Proof. Fix w1 , w2 ∈ H and c1 , c2 ∈ R. By (2.3.7), the maps
w 7→ Re[BQ (f (c1 w1 + c2 w2 ), f (w))] and w 7→ Re[BQ (c1 f (w1 ) + c2 f (w2 ), f (w))]
are identical. By the surjectivity of f together with the Riesz representation theorem, this implies that f (c1 w1 + c2 w2 ) = c1 f (w1 ) + c2 f (w2 ).
⊳
Claim 2.3.8. f preserves F-lines.
Proof. For each x ∈ H \ {0}, the F-line xF may be defined using the quantity

|BQ | via the formula

xF = {y ∈ H : ∀w ∈ H, |BQ (x, w)| = 0 ⇔ |BQ (y, w)| = 0} .
The claim therefore follows from (2.3.8).

⊳

12

2. ALGEBRAIC HYPERBOLIC SPACES

From Claim 2.3.8, we see that for all x ∈ H \ {0} and a ∈ F, there exists

σx (a) ∈ F such that

f (xa) = f (x)σx (a).

Claim 2.3.9. For x, y ∈ H \ {0},
σx (a) = σy (a).
Proof. By Claim 2.3.7,
[f (x) + f (y)]σx+y (a) = f (xa + ya) = f (x)σx (a) + f (y)σy (a).
Rearranging, we see that
f (x)[σx+y (a) − σx (a)] + f (y)[σx+y (a) − σy (a)] = 0.
If x and y are linearly independent, then σx+y (a) − σx (a) = 0 and σx+y (a) −

σy (a) = 0, so σx (a) = σy (a). But the general case clearly follows from the linearly
independent case.

⊳

For a ∈ F, denote the common value of σx (a) (x ∈ H \ {0}) by σ(a). Then
(2.3.9)

f (xa) = f (x)σ(a) ∀x ∈ H ∀a ∈ F.

Claim 2.3.10. σ ∈ Aut(F).
Proof. The R-linearity of σ follows from Claim 2.3.7, and the bijectivity of σ
follows from the bijectivity of f . Fix x ∈ H \ {0} arbitrary. For a, b ∈ F,
f (x)σ(ab) = f (xab) = f (xa)σ(b) = f (x)σ(a)σ(b),
which proves that σ is a multiplicative homomorphism.

⊳

Thus f ∈ O∗F (H; E), and so T = f ⊕I ∈ O∗F (L; Q). But [T ] = g by (2.3.5), so g ∈

PO∗F (L; Q). This completes the first part of the proof, namely that PO∗F (L; Q) =
Isom(H).
To complete the proof, we need to show that POF (L; Q) is equal to PO∗F (L; Q) if
F = R, Q and is of index 2 in PO∗F (L; Q) if F = C. If F = R, this is obvious. If F = C,

it follows from the semidirect product structure O∗F (L; Q) = OF (L; Q) ⋉ {σ J : σ ∈
Aut(F)} together with the fact that Aut(F) = {I, z 7→ z̄} ≡ Z2 .

If F = Q, then Aut(F) = {Φa : a ∈ S(Q)}, where Φa (b) = aba−1 . Here
S(F) = {a ∈ F : |a| = 1}. So O∗Q (L; Q) 6= OQ (L; Q); nevertheless, we will show

that PO∗Q (L; Q) = POQ (L; Q). Fix [T ] ∈ PO∗Q (L; Q), and fix a ∈ S(Q) for which

σT = Φa . Consider the map
(2.3.10)

Ta (x) = xa.

2.3. ISOMETRIES OF ALGEBRAIC HYPERBOLIC SPACES

13

−1
= I, so Ta T is
We have Ta ∈ O∗Q (L; Q) and σTa = Φ−1
a . Thus σTa T = Φa Φa

F-linear. But

[Ta T ] = [T ],
so [T ] ∈ POQ (L; Q). The completes the proof of Theorem 2.3.3.



Remark 2.3.11. Using algebraic language, the automorphisms Φa of Q are
inner automorphisms, while the automorphism z 7→ z̄ of C is an outer automorphism. Although both inner and outer automorphisms contribute to the quotient O∗F (L; Q)/ OF (L; Q), only the outer automorphisms contribute to the quotient

PO∗F (L; Q)/ POF (L; Q). This explains why the index #(PO∗F (L; Q)/ POF (L; Q)) is
smaller when F = Q than when F = C: although the group Aut(Q) is much larger
than Aut(C), it consists entirely of inner automorphisms, while Aut(C) has an outer
automorphism.
Definition 2.3.12. The bordification of H is its closure relative to the topological space P(L), i.e.
bord H = {[x] : Q(x) ≤ 0}.
The boundary of H is its topological boundary relative to P(L), i.e.
∂H = bord H \ H = {[x] : Q(x) = 0}.
The following is a corollary of Theorem 2.3.3:
Corollary 2.3.13. Every isometry of H extends uniquely to a homeomorphism
of bord H.
Proof. If T ∈ O∗F (L; Q), then the formula (2.3.1) defines a homeomorphism

of bord H which extends the action of [T ] on H. The uniqueness is automatic.



Remark 2.3.14. Corollary 2.3.13 can also be proven independently of Theorem
2.3.3 via the theory of hyperbolic metric spaces; cf. Lemma 3.4.25 and Proposition
3.5.3.
The following observation will be useful in the sequel:
Observation 2.3.15. Fix [x], [y] ∈ bord H. Then
BQ (x, y) = 0 ⇔ [x] = [y] ∈ ∂H.
Proof. If either [x] or [y] is in H, this follows from Lemma 2.2.4. Suppose that
[x], [y] ∈ ∂H, and that BQ (x, y) = 0. Then Q is identically zero on xF + yF. Thus

(xF+ yF)∩H = {0}, and so xF+ yF is one-dimensional. This implies [x] = [y].



14

2. ALGEBRAIC HYPERBOLIC SPACES

2.4. Totally geodesic subsets of algebraic hyperbolic spaces
Given two pairs (X, bord X) and (Y, bord Y ), where X and Y are metric spaces
contained in the topological spaces bord X and bord Y (and dense in these spaces),
an isomorphism between (X, bord X) and (Y, bord Y ) is a homeomorphism between
bord X and bord Y which restricts to an isometry between X and Y .
Proposition 2.4.1. Let K ≤ F be an R-subalgebra, and let V ≤ L be a closed

(right) K-module such that

BQ (x, y) ∈ K ∀x, y ∈ V.

(2.4.1)

Then either [V ] ∩ H =  and #([V ] ∩ bord H) ≤ 1, or ([V ] ∩ H, [V ] ∩ bord H) is
isomorphic to an algebraic hyperbolic space together with its closure.
Proof.
Case 1: [V ] ∩ H 6= . In this case, fix [z] ∈ [V ] ∩ H, and let z be a representative
of [z] with Q(z) = −1. By Lemma 2.2.4, Q is positive-definite on z⊥ .
We leave it as an exercise that the quadratic forms Q ↿ z⊥ and E ↿ z⊥
agree up to a bounded multiplicative error factor, which implies that z⊥
√
is complete with respect to the norm Q.
From (2.4.1), we see that (V ∩ z⊥ , BQ ) is a K-Hilbert space. By

the usual Gram–Schmidt process, we may construct an orthonormal basis
′
(ei )i∈J ′ for V ∩z⊥ , thus proving that V ∩z⊥ is isomorphic to HKJ for some
J ′ ∪{0}

set J ′ . Thus V is isomorphic to LK
′
′
(HJK , bord HJK ).

, and so ([V ] ∩ H, [V ] ∩ bord H) is

isomorphic to
Case 2: [V ] ∩ H = . We need to show that #([V ] ∩ bord H) ≤ 1. By contradiction fix [x], [y] ∈ [V ] distinct, and let x, y ∈ V be representatives. By
Observation 2.3.15, BQ (x, y) 6= 0. On the other hand, Q(x) = Q(y) = 0

since [x], [y] ∈ ∂H. Thus Q(x − yB(x, y)−1 ) = −2 < 0. On the other
hand, x − yB(x, y)−1 ∈ V by (2.4.1). Thus [x − yB(x, y)−1 ] ∈ [V ] ∩ H,

a contradiction.


Definition 2.4.2. A totally geodesic subset of an algebraic hyperbolic space
H is a set of the form [V ] ∩ bord H, where V is as in Proposition 2.4.1. A totally

geodesic subset is nontrivial if it contains an element of H.

Remark 2.4.3. As with Definition 2.2.5, the terminology “totally geodesic” is
motivated here by the finite-dimensional situation, where totally geodesic subsets
correspond precisely with the closures of those submanifolds which are totally geodesic in the sense of Riemannian geometry; see [147, Proposition A.4 and A.7].

2.4. TOTALLY GEODESIC SUBSETS OF ALGEBRAIC HYPERBOLIC SPACES

15

However, note that we consider both the empty set and singletons in ∂H to be
totally geodesic.
Remark 2.4.4. If V ≤ L is a closed K-module satisfying (2.4.1), then for each
a ∈ F \ {0}, V a is a closed a−1 Ka-module satisfying (2.4.1) (with K = a−1 Ka).
Lemma 2.4.5. The intersection of any collection of totally geodesic sets is totally geodesic.
Proof. Suppose that (Sα )α∈A is a collection of totally geodesic sets, and supT
pose that S = α Sα 6= . Fix [z] ∈ S, and let z be a representative of [z]. Then for

each α ∈ A, there exist (cf. Remark 2.4.4) an R-algebra Kα and a closed Kα -subspace
Vα ≤ L satisfying (2.4.1) (with K = Kα ) such that z ∈ Vα and Sα = [Vα ] ∩ bord H.
T
T
Let K = α Kα and V = α Vα . Clearly, V is a K-module and satisfies (2.4.1).
We have [V ] ∩ bord H ⊆ S. To complete the proof, we must show the converse
direction. Fix [x] ∈ S \ {[z]}. By Observation 2.3.15, there exists a representative
x of [x] such that BQ (z, x) = 1. Then for each α, we may find aα ∈ F \ {0} such

that xaα ∈ Vα . We have

aα = BQ (z, x)aα = BQ (z, xaα ) ∈ Kα .
Since Vα is a Kα -module, this implies x ∈ Vα . Since α was arbitrary, x ∈ V , and
so [x] ∈ [V ] ∩ bord H.

Remark 2.4.6. Given K ⊆ bord H, Lemma 2.4.5 implies that there exists a
smallest totally geodesic set containing K. If we are only interested in the geometry
of K, then by Proposition 2.4.1 we can assume that this totally geodesic set is really
our ambient space. In such a situation, we may without loss of generality suppose
that there is no proper totally geodesic subset of bord H which contains K. In this
case we say that K is irreducible.
Warning. Although the intersection of any collection of totally geodesic sets
is totally geodesic, it is not necessarily the case that the decreasing intersection of
nontrivial totally geodesic sets is nontrivial; cf. Remark 11.2.19.
The main reason that totally geodesic sets are relevant to our development is
their relationship with the group of isometries. Specifically, we have the following:
Theorem 2.4.7. Let (gn )∞
1 be a sequence in Isom(H), and let
n
o
(2.4.2)
S = [x] ∈ bord H : gn ([x]) −
→ [x] .
n

Then either S ⊆ ∂H and #(S) = 2, or S is a totally geodesic set.

16

2. ALGEBRAIC HYPERBOLIC SPACES

Remark 2.4.8. An important example is the case where the sequence (gn )∞
1
is constant, say gn = g for all n. Then S is precisely the fixed point set of g:
S = Fix(g) := {[x] ∈ bord H : g([x]) = [x]} .
If H is finite-dimensional, then it is possible to reduce Theorem 2.4.7 to this special
case by a compactness argument.
Proof of Theorem 2.4.7. If S = , then the statement is trivial. Suppose
that S 6= , and fix [z] ∈ S.

Step 1: Choosing representatives Tn . From the proof of Theorem 2.3.3, we see
that each gn may be written in the form [Tn ] for some Tn ∈ O∗F (L; Q). We have

some freedom in choosing the representatives Tn ; specifically, given an ∈ S(F) we
may replace Tn by Tn Tan , where Tan is defined by (2.3.10).
Since gn ([z]) → [z], there exist representatives zn of gn ([z]) such that zn → z.
For each n, there is a unique representative Tn of gn such that
(Tn z)cn = zn for some cn ∈ R \ {0}.
Then
(Tn z)cn → z.
Remark 2.4.9. If F = Q, it may be necessary to choose Tn ∈ O∗F (L; Q) \

OF (L; Q), despite the fact that each gn can be represented by an element of
OF (L; Q).
Step 2: A totally geodesic set. Write σn = σTn , and let
K = {a ∈ F : σn (a) → a}
n
o
V = x ∈ L : Tn x −
→x .
n

Then K is an R-subalgebra of F, and V is a K-module. Given x, y ∈ V , by Observation 2.3.6 we have
σn (BQ (x, y)) = BQ (Tn x, Tn y) −
→ BQ (x, y),
n

so B(x, y) ∈ K. Thus V satisfies (2.4.1). If V is closed, then the above observations
show that [V ] ∩ bord H is totally geodesic. However, this issue is a bit delicate:
Claim 2.4.10. If #([V ] ∩ bord H) ≥ 2, then V is closed.
Proof. Suppose that #([V ] ∩ bord H) ≥ 2. The proof of Proposition 2.4.1
shows that [V ] ∩ H 6= . Thus, there exists x ∈ V for which [x] ∈ H. In particular,
gn ([x]) → [x]. Letting o = [(1, 0)], we have

dH (o, gn (o)) ≤ 2dH (o, [x]) + dH ([x], gn ([x])) −
→ 2dH (o, [x]).
n

2.4. TOTALLY GEODESIC SUBSETS OF ALGEBRAIC HYPERBOLIC SPACES

17

In particular dH (o, gn (o)) is bounded, say dH (o, gn (o)) ≤ C.
Lemma 2.4.11. Fix T ∈ O∗F (L; Q), and let kT k denote the operator norm of T .
Then
kT k = edH (o,[T ](o)) .
Proof. Write T = Tj,t (A ⊕ I), where Tj,t is a Lorentz boost (cf. (2.3.3)) and
A ∈ O∗F (H; E). Then
[T ](o) = [Tj,t ](o) = [(cosh(t), sinh(t), 0)].
Here the second entry represents the jth coordinate. In particular,
cosh(t)
|BQ ((1, 0), (cosh(t), sinh(t), 0))|
= cosh(t).
=
cosh dH (o, [T ](o)) = p
1
|Q(1, 0)| · |Q(cosh(t), sinh(t), 0)|

On the other hand,



cosh(t)


kT k = kTj,t k =  sinh(t)



sinh(t)
cosh(t)
I

This completes the proof.


t
 =e .

⊳

Thus kTn k ≤ eC for all n, and so the sequence (Tn )∞
1 is equicontinuous. It follows
that V is closed.

⊳

Since #([V ] ∩ bord H) ≤ 1 implies that [V ] ∩ bord H is totally geodesic, we conclude
that [V ] ∩ bord H is totally geodesic, regardless of whether or not V is closed.

Remark 2.4.12. When #([V ] ∩ bord H) ≤ 1, there seems to be no reason to
think that V should be closed.
Step 3: Relating S to [V ] ∩ bord H. The object of this step is to show that
S = [V ] ∩ bord H unless S ⊆ ∂H and #(S) ≤ 2. For each [x] ∈ S \ {[z]}, let x

be a representative of [x] such that BQ (z, x) = 1; this is possible by Observation
([x])
2.3.15. It is possible to choose a sequence of scalars (an )∞
n=1 in F \ {0} such that
([x])

(Tn x)an

([z])

→ x. Let an

= cn . For [x], [y] ∈ S, we have

(2.4.3)
([x])

an

([x])

= an
σTn (BQ (x, y))a([y])
n

BQ (Tn x, Tn y)a([y])
n

(by Observation 2.3.6)

= BQ ((Tn x)a([x])
, (Tn y)a([y])
)
n
n
−
→ BQ (x, y).
n

In particular,
(2.4.4)

|a([x])
| · |a([y])
|−
→ 1 whenever BQ (x, y) 6= 0.
n
n
n

18

2. ALGEBRAIC HYPERBOLIC SPACES

Claim 2.4.13. Unless S ⊆ ∂H and #(S) ≤ 2, then for all [x] ∈ S we have
|a([x])
|−
→ 1.
n

(2.4.5)

n

Proof. We first observe that it suffices to demonstrate (2.4.5) for one value
of x; if (2.4.5) holds for x and [y] 6= [x], then BQ (x, y) 6= 0 by Observation 2.3.15
([y])
and so (2.4.4) implies |an | → 1.
Now suppose that S * ∂H, and choose [x] ∈ S ∩ H. Then BQ (x, x) 6= 0, and

so (2.4.4) implies (2.4.5).
Finally, suppose that #(S) ≥ 3, and choose [x], [y], [z] ∈ S distinct. By (2.4.4)
([x])

([y])

([x])

([z])

together with Observation 2.3.15, we have |an | · |an | → 1, |an | · |an | → 1,
([y])
([z])
and |an | · |an | → 1. Multiplying the first two formulas and dividing by the
([x])

third, we see that |an

| → 1.

⊳

For the remainder of the proof we assume that either S * ∂H or #(S) ≥ 3.
Plugging z = x into (2.4.5), we see that cn → 1. In particular, [z] ∈ [V ]∩bord H.

Now fix [x] ∈ S \ {[z]}. Since cn → 1 and BQ (z, x) = 1, (2.4.3) becomes
a([x])
→ 1.
n
Thus x ∈ V , and so [x] ∈ [V ] ∩ bord H.



2.5. Other models of hyperbolic geometry
Fix F ∈ {R, C, Q} and a set J, and let H = HJF . The pair (H, bord H) is known
as the hyperboloid model of hyperbolic geometry (over the division algebra F and
in dimension #(J)). In this section we discuss two other important models of
hyperbolic geometry. Note that the Poincaré ball model, which many of the figures
of later chapters are drawn in, is not discussed here. References for this section
include [45, 78].
2.5.1. The (Klein) ball model. Let
B = BJF = {x ∈ H := HFJ : kxk < 1},
and let bord B denote the closure of B relative to H.
Observation 2.5.1. The map eB,H : bord B → bord H defined by the equation
eB,H (x) = [(1, x)]
is a homeomorphism, and eB,H (B) = H. Thus if we let
(2.5.1)

|1 − BE (x, y)|
p
,
cosh dB (x, y) = cosh dH (eB,H (x), eB,H (y)) = p
1 − kxk2 1 − kyk2

2.5. OTHER MODELS OF HYPERBOLIC GEOMETRY

19

then eB,H is an isomorphism between (B, bord B) and (H, bord H).
The pair (B, bord B) is called the ball model of hyperbolic geometry. It is often
convenient for computations, especially those for which a single point plays an
important role: by Observation 2.3.2, such a point can be moved to the origin
0 ∈ B via an isomorphism of (B, bord B).
Remark 2.5.2. We should warn that the ball model BJR of real hyperbolic
geometry is not the same as the well-known Poincaré model, rather, it is the same
as the Klein model.

0.

Observation 2.5.3. For all T ∈ O∗F (H; E), T ↿ B is an isometry which stabilizes
Proposition 2.5.4. In fact,
Stab(Isom(B); 0) = {T ↿ B : T ∈ O∗F (H; E)}.
Proof. This is an immediate consequence of Theorem 2.3.3.



2.5.2. The half-space model. Now suppose F = R.7 Assume that 1 ∈ J,
and let

E = EJ = x ∈ H := HFJ x1 > 0 .
We will view E as resting inside the larger space

b := H ∪ {∞}.
H

b is defined as follows: a subset U ⊆ H
b is open if and only if
The topology on H
U ∩ H is open and (∞ ∈ U ⇒ H \ U is bounded).

b according to the topology
The boundary and closure of E will be subsets of H
defined above, i.e.
∂E = {x ∈ H : x1 = 0} ∪ {∞}
bord E = {x ∈ H : x1 ≥ 0} ∪ {∞}.
Proposition 2.5.5. The map eE,H : bord E → bord H defined by the formula






2x
i
=
6
0,
1




 i








 x 6= ∞
 1 + kxk2 i = 0 






(2.5.2)
eE,H (x) =


2

1
−
kxk
i
=
1


i∈J∪{0}



[(1, −1, 0)]
x=∞

7The appropriate analogue of the half-space model when F ∈ {C, Q} is the paraboloid model ; see

e.g. [78, Chapter 4].

20

2. ALGEBRAIC HYPERBOLIC SPACES

is a homeomorphism, and eE,H (E) = H. Thus if we let
(2.5.3)

cosh dE (x, y) = cosh dH (eE,H (x), eE,H (y)) = 1 +

ky − xk2
,
2x1 y1

then eE,H is an isomorphism between (E, bord E) and (H, bord H).
Proof. For x ∈ bord E \ {∞},
Q(eE,H (x)) = −(1 + kxk2 )2 + (1 − kxk2 )2 +

X

i∈J\{1}

(2xi )2 = −4x21 .

It follows that eE,H (E) ⊆ H and eE,H (∂E) ⊆ ∂H. Calculation verifies that the map


x /2


i
=
6
1
i


 p
if x0 + x1 = 2
 −Q(x)/2 i = 1
(2.5.4)
eH,E ([x]) =
i∈J



∞
if x = (1, −1, 0)

is both a left and a right inverse of eE,H . Notice that it is defined in a way such
that for each [x] ∈ bord H, there is a unique representative x of [x] for which the
formula (2.5.4) makes sense. We leave it to the reader to verify that eE,H and eH,E
are both continuous, and that (2.5.3) holds.



The point ∞ ∈ ∂E, corresponding to the point [(1, −1, 0)] ∈ ∂H, plays a special

role in the half-space model. In fact, the half-space model can be thought of as an
attempt to understand the geometry of hyperbolic space when a single point on the
boundary is fixed. Consequently, we are less interested in the set of all isometries
of E than simply the set of all isometries which fix ∞.
J\{1}

Observation 2.5.6 (Poincaré extension). Let B = ∂E \ {∞} = HR

, and

let g : B → B be a similarity, i.e. a map of the form
g(x) = λT x + b,

where λ > 0, T ∈ OR (B; E), and b ∈ B. Then the map b
g : bord E → bord E defined
by the formula

(λx , g(π(x))) x 6= ∞
1
(2.5.5)
g (x) =
b
∞
x=∞

is an isomorphism of (E, bord E); in particular, gb ↿ E is an isometry of E. Here

π : H → B is the natural projection.

Proof. This is immediate from (2.5.3).

The isometry gb defined by (2.5.5) is called the Poincaré extension of g to E.



Remark 2.5.7. Intuitively we shall think of the number x1 as representing the
height of a point x ∈ bord E. Then (2.5.5) says that if g : B → B is an isometry,

2.5. OTHER MODELS OF HYPERBOLIC GEOMETRY

21

then the Poincaré extension of g is an isometry of E which preserves the heights of
points.
Proposition 2.5.8. For all g ∈ Isom(E) such that g(∞) = ∞, there exists a
h.
similarity h : B → B such that g = b

Proof. By Theorem 2.3.3, there exists T ∈ O(L; Q) such that [T ] = eE,H ◦ g ◦
e−1
.
E,H This gives an explicit formula for g, and one must check that if [T ] preserves
[(1, −1, 0)], then g is a Poincaré extension.



2.5.3. Transitivity of the action of Isom(H) on ∂H. Using the ball and
half-plane models of hyperbolic geometry, it becomes easy to prove the following
assertion:
Proposition 2.5.9. If F = R, the group Isom(H) acts triply transitively on ∂H.
This complements the fact that Isom(H) acts transitively on H (Observation
2.3.2).
Proof. By Observation 2.5.1 and Proposition 2.5.5, we may switch between
models as convenient. It is clear that Isom(B) acts transitively on ∂B, and that
Stab(Isom(E); ∞) acts doubly transitively on ∂E \ {∞}. Therefore given any triple
(ξ1 , ξ2 , ξ3 ), we may conjugate to B, conjugate ξ1 to a standard point, conjugate to
E while conjugating ξ1 to ∞, and then conjugate ξ2 , ξ3 to standard points.



We end this chapter with a convention:
J
Convention 6. When α is a cardinal number, Hα
F will denote HF for any set

J of cardinality α, but particularly J = {1, . . . , n} if α = n ∈ N and J = N if α =
#(N)
= HN
#(N). Moreover, H∞
F , the unique (up to
F will always be used to denote HF
isomorphism) infinite-dimensional separable algebraic hyperbolic space defined over
F. Finally, real hyperbolic spaces will be denoted without using R as a subscript,
J
J
α
α
e.g. H∞ = H∞
R , B = BR , H = HR .

CHAPTER 3

R-trees, CAT(-1) spaces, and Gromov hyperbolic
metric spaces
In this chapter we review the theory of “negative curvature” in general metric
spaces. A good reference for this subject is [39]. We begin by defining the class of Rtrees, the main class of examples we will talk about in this monograph other than the
class of algebraic hyperbolic spaces, which we will discuss in more detail in Chapter
14. Next we will define CAT(-1) spaces, which are geodesic metric spaces whose
triangles are “thinner” than the corresponding triangles in two-dimensional real
hyperbolic space H2 . Both algebraic hyperbolic spaces and R-trees are examples of
CAT(-1) spaces. The next level of generality considers Gromov hyperbolic metric
spaces. After defining these spaces, we proceed to define the boundary ∂X of a
hyperbolic metric space X, introducing the families of so-called visual metametrics
and extended visual metrics on the bordification bord X := X ∪ ∂X. We show that
the bordification of an algebraic hyperbolic space X is isomorphic to its closure
bord X defined in Chapter 2; under this isomorphism, the visual metric on ∂Bα is
proportional to the Euclidean metric.

3.1. Graphs and R-trees
To motivate the definition of R-trees we begin by defining simplicial trees, which
requires first defining graphs.
Definition 3.1.1. A weighted undirected graph is a triple (V, E, ℓ), where V
is a nonempty set, E ⊆ V × V \ {(x, x) : x ∈ V } is invariant under the map
(x, y) 7→ (y, x), and ℓ : E → (0, ∞) is also invariant under (x, y) → (y, x). (If ℓ ≡ 1,
the graph is called unweighted, and can be denoted simply (V, E).) The graph is
called connected if for all x, y ∈ V , there exist x = z0 , z1 , . . . , zn = y such that

(zi , zi+1 ) ∈ E for all i = 0, . . . , n − 1. If (V, E, ℓ) is connected, then the path metric
on V is the metric
)
(n−1
z0 = x, zn = y,
X
ℓ(zi , zi+1 )
(3.1.1)
dE,ℓ (x, y) := inf
(zi , zi+1 ) ∈ E ∀i = 0, . . . , n − 1
i=0
23

24

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

The geometric realization of the graph (V, E, ℓ) is the metric space


[
[0, ℓ(v, w)] / ∼,
X = X(V, E, ℓ) = V ∪
(v,w)∈E

where ∼ represents the following identifications:
v ∼ ((v, w), 0) ∀(v, w) ∈ E

((v, w), t) ∼ ((w, v), ℓ(v, w) − t) ∀(v, w) ∈ E ∀t ∈ [0, ℓ(v, w)]
and the metric d on X is given by

d ((v0 , v1 ), t), ((w0 , w1 ), s) = min {|t − iℓ(v0 , v1 )| + d(vi , wj ) + |s − jℓ(w0 , w1 )|}.
i∈{0,1}
j∈{0,1}

(The geometric realization of a graph is sometimes also called a graph. In the
sequel, we shall call it a geometric graph.)
Example 3.1.2 (The Cayley graph of a group). Let Γ be a group, and let
E0 ⊆ Γ be a generating set. (In most circumstances E0 will be finite; there is an

exception in Example 13.3.2 below.) Assume that E0 = E0−1 . The Cayley graph of
Γ with respect to the generating set E0 is the unweighted graph (Γ, E), where
(3.1.2)

(γ, β) ∈ E ⇔ γ −1 β ∈ E0 .

More generally, if ℓ0 : E0 → (0, ∞) satisfies ℓ0 (g −1 ) = ℓ0 (g), the weighted Cayley

graph of Γ with respect to the pair (E0 , ℓ0 ) is the graph (Γ, E, ℓ), where E is defined
by (3.1.2), and
(3.1.3)

ℓ(γ, β) = ℓ0 (γ −1 β).

The path metric of a Cayley graph is called a Cayley metric.
Remark 3.1.3. The equations (3.1.2), (3.1.3) guarantee that for each γ ∈ Γ,
the map Γ ∋ β → γβ ∈ Γ is an isometry of Γ with respect to any Cayley metric.

This isometry extends in a unique way to an isometry of the geometric Cayley
graph X = X(Γ, E, ℓ). The map sending γ to this isometry is a homomorphism
from Γ to Isom(X), and is called the natural action of Γ on X.
Remark 3.1.4. The path metric (3.1.1) satisfies the following universal prop-

erty: If Y is a metric space and if φ : V → Y satisfies d(φ(v), φ(w)) ≤ ℓ(v, w) for
every (v, w) ∈ E, then d(φ(v), φ(w)) ≤ d(v, w) for every v, w ∈ V .
Remark 3.1.5. The main difference between the metric space (V, dE,ℓ ) and the
geometric graph X = X(V, E, ℓ) is that the latter is a geodesic metric space. A
metric space X is said to be geodesic if for every p, q ∈ X, there exists an isometric
embedding π : [t, s] → X such that π(t) = p and π(s) = q, for some t, s ∈ R. The

3.1. GRAPHS AND R-TREES

25

set π([t, s]) is denoted [p, q] and is called a geodesic segment connecting p and q.
The map π is called a parameterization of the geodesic segment [p, q]. (Note that
although [q, p] = [p, q], π is not a parameterization of [q, p].)
Warning: Although we may denote any geodesic segment connecting p and q
by [p, q], such a geodesic segment is not necessarily unique. A geodesic metric space
X is called uniquely geodesic if for every p, q ∈ X, the geodesic segment connecting
p and q is unique.
Notation 3.1.6. If π : [0, t0 ] → X is a parameterization of the geodesic seg-

ment [p, q], then for each t ∈ [0, t0 ], [p, q]t denotes the point π(t), i.e. the unique
point on the geodesic segment [p, q] such that d(p, [p, q]t ) = t.

We are now ready to define the class of simplicial trees. Let (V, E, ℓ) be a
weighted undirected graph. A cycle in (V, E, ℓ) is a finite sequence of distinct
vertices v1 , . . . , vn ∈ V , with n ≥ 3, such that
(3.1.4)

(v1 , v2 ), (v2 , v3 ), . . . , (vn−1 , vn ), (vn , v1 ) ∈ E.

Definition 3.1.7. A simplicial tree is the geometric realization of a weighted
undirected graph with no cycles. A Z-tree (or unweighted simplicial tree, or just
tree 1) is the geometric realization of an unweighted undirected graph with no cycles.
Example 3.1.8. Let F2 (Z) denote the free group on two elements γ1 , γ2 . Let
E0 = {γ1 , γ1−1 , γ2 , γ2−1 }. The geometric Cayley graph of F2 (Z) with respect to the
generating set E0 is an unweighted simplicial tree.
Example 3.1.9. Let V = {C, p, q, r}, and fix ℓp , ℓq , ℓr > 0. Let
E = {(C, x), (x, C) : x = p, q, r},
ℓ(C, x) = ℓ(x, C) = ℓx .
The geometric realization of the graph (V, E, ℓ) is a simplicial tree; see Figure
3.1.1. It will be denoted ∆ = ∆(p, q, r), and will be called a tree triangle. For
x, y ∈ {p, q, r} distinct, the distance between x and y is given by
d(x, y) = ℓx + ℓy .
Solving for ℓp in terms of d(p, q), d(p, r), d(q, r) gives
(3.1.5)

ℓp = d(p, C) =

1
[d(p, q) + d(p, r) − d(q, r)].
2

Definition 3.1.10. A metric space X is an R-tree if for all p, q, r ∈ X, there

exists a tree triangle ∆ = ∆(p, q, r) and an isometric embedding ι : ∆ → X sending
p, q, r to p, q, r, respectively.
1However, in [167], the word “trees” is used to refer to what are now known as R-trees.

26

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

Definition 3.1.11. Let X be an R-tree, fix p, q, r ∈ X, and let ι : ∆ → X

be as above. The point C = C(p, q, r) := ι(C) is called the center of the geodesic
triangle ∆ = ∆(p, q, r).
As the name suggests, every simplicial tree is an R-tree; the converse does not
hold; see e.g. [51, Example on p.50]. Before we can prove that every simplicial tree
is an R-tree, we will need a lemma:
Lemma 3.1.12 (Cf. [51, p.29]). Let X be a metric space. The following are
equivalent:
(A) X is an R-tree.
(B) There exists a collection of geodesics G, with the following properties:
(BI) For each x, y ∈ X, there is a geodesic [x, y] ∈ G connecting x and y.

(BII) Given [x, y] ∈ G and z, w ∈ [x, y], we have [z, w] ∈ G, where [z, w] is
interpreted as the set of points in [x, y] which lie between z and w.

(BIII) Given x1 , x2 , x3 ∈ X distinct and geodesics [x1 , x2 ], [x1 , x3 ], [x2 , x3 ] ∈
G, at least one pair of the geodesics [xi , xj ], i 6= j, has a nontrivial

intersection. More precisely, there exist distinct i, j, k ∈ {1, 2, 3} such
that
[xi , xj ] ∩ [xi , xk ] % {xi }.

Proof of (A) ⇒ (B). Note that (BI) and (BII) are true for any uniquely
geodesic metric space. Given x1 , x2 , x3 distinct, let C be the center. Then xi 6= C
for some i; without loss of generality x1 6= C. Then

[x1 , x2 ] ∩ [x1 , x3 ] = [x1 , C] % {x1 }.

Proof of (B) ⇒ (A). We first show that given points x1 , x2 , x3 ∈ X and
T
geodesics [x1 , x2 ], [x1 , x3 ], [x2 , x3 ] ∈ G, the intersection i6=j [xi , xj ] is nonempty.

Indeed, suppose not. For i = 2, 3 let γi : [0, d(x1 , xi )] → X be a parameterization
of [x1 , xi ], and let

t1 = max{t ≥ 0 : γ2 (t) = γ3 (t)}.
By replacing x with γ2 (t1 ) = γ3 (t1 ) and using (BII), we may without loss of generality assume that t1 = 0, or equivalently that [x1 , x2 ] ∩ [x1 , x3 ] = {x1 }. Simi-

larly, we may without loss of generality assume that [x2 , x1 ] ∩ [x2 , x3 ] = {x2 } and
[x3 , x1 ] ∩ [x3 , x2 ] = {x3 }. But then (BIII) implies that x1 , x2 , x3 cannot be all
T
distinct. This immediately implies that i6=j [xi , xj ] 6= .
To complete the proof, we must show that X is uniquely geodesic. Indeed
suppose that for some x1 , x2 ∈ X, there is more than one geodesic connecting x1

and x2 . Let [x1 , x2 ] ∈ G be a geodesic connecting x1 and x2 , and let [x1 , x2 ]′ be

3.2. CAT(-1) SPACES

27

q

C(p, q, r)

r

p

Figure 3.1.1. A geodesic triangle in an R-tree
another geodesic connecting x1 and x2 . Then there exists x3 ∈ [x1 , x2 ]′ \ [x1 , x2 ].
T
By the above paragraph, there exists w ∈ i6=j [xi , xj ]. Since w ∈ [xi , x3 ], we have
d(xi , w) ≤ d(xi , x3 ).

(3.1.6)

On the other hand, since w ∈ [x1 , x2 ] and x3 ∈ [x1 , x2 ]′ , we have
d(x1 , x2 ) = d(x1 , w) + d(x2 , w) ≤ d(x1 , x3 ) + d(x2 , x3 ) = d(x1 , x2 ).
It follows that equality holds in (3.1.6), i.e. d(xi , w) = d(xi , x3 ). Since w ∈ [xi , x3 ],
this implies w = x3 . But then x3 = w ∈ [x1 , x2 ], a contradiction.

Corollary 3.1.13. Every simplicial tree is an R-tree.
Proof. Let X = X(V, E, ℓ) be a simplicial tree, and let G be the collection
of all geodesics; then (BI) and (BII) both hold. By contradiction, suppose that
there exist points x1 , x2 , x3 ∈ X such that [xi , xj ] ∩ [xi , xk ] = {xi } for all distinct
i, j, k ∈ {1, 2, 3}. Then the path [x1 , x2 ] ∪ [x2 , x3 ] ∪ [x3 , x1 ] is equal to the union of
the edges of a cycle of the graph (V, E, ℓ). This is a contradiction.



We shall investigate R-trees in more detail in Chapter 14, where we will give
various examples of R-trees together with groups acting isometrically on them.
3.2. CAT(-1) spaces
The following definitions have been modified from [39, p.158], to which the
reader is referred for more details.
A geodesic triangle in X consists of three points p, q, r ∈ X (the vertices of

the triangle) together with a choice of three geodesic segments [p, q], [q, r], and
[r, p] joining them (the sides). Such a geodesic triangle will be denoted ∆(p, q, r),
although we note that this could cause ambiguity if X is not uniquely geodesic.
Although formally ∆(p, q, r) is an element of X 3 ×P(X)3 , we will sometimes identify
∆(p, q, r) with the set [p, q] ∪ [q, r] ∩ [r, p] ⊆ X, writing x ∈ ∆(p, q, r) if x ∈ [p, q] ∪

[q, r] ∪ [r, p].

28

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

A triangle ∆ = ∆(p, q, r) in H2 is called a comparison triangle for ∆ = ∆(p, q, r)
if d(p, q) = d(p, q), d(q, r) = d(q, r), and d(p, r) = d(p, r). Any triangle admits a
comparison triangle, unique up to isometry. For any point x ∈ [p, q], we define its

comparison point x ∈ [p, q] to be the unique point such that d(x, p) = d(x, p) and
d(x, q) = d(x, q). In the notation above, the comparison point of [p, q]t is equal to
[p, q]t for all t ∈ [0, d(p, q)] = [0, d(p, q)]. For x ∈ [q, r] and x ∈ [r, p], the comparison
point is defined similarly.
Let X be a metric space and let ∆ be a geodesic triangle in X. We say that ∆
satisfies the CAT(-1) inequality if for all x, y ∈ ∆,
(3.2.1)

d(x, y) ≤ d(x, y),

where x and y are any2 comparison points for x and y, respectively. Intuitively, ∆
satisfies the CAT(-1) inequality if it is “thinner” than its comparison triangle ∆.
Definition 3.2.1. X is a CAT(-1) space if it is a geodesic metric space and if
all of its geodesic triangles satisfy the CAT(-1) inequality.
Observation 3.2.2 ([39, Proposition II.1.4(1)]). CAT(-1) spaces are uniquely
geodesic.
Proof. Let X be a CAT(-1) space, and suppose that two points p, q ∈ X are

connected by two geodesic segments [p, q] and [p, q]′ . Fix t ∈ [0, d(p, q)] and let
x = [p, q]t , x′ = [p, q]′t . Consider the triangle ∆(p, q, x) determined by the geodesic

segments [p, q]′ , [p, x], and [x, q], and a comparison triangle ∆(p, q, x). Then x and
x′ have the same comparison point x, so by the CAT(-1) inequality
d(x, x′ ) ≤ d(x, x) = 0,
and thus x = x′ . Since t was arbitrary, it follows that [p, q] = [p, q]′ . Since [p, q]′
was arbitrary, [p, q] is the unique geodesic segment connecting p and q.



3.2.1. Examples of CAT(-1) spaces. In this text we concentrate on two
main examples of CAT(-1) spaces: algebraic hyperbolic spaces and R-trees. We
therefore begin by proving the following result which implies that algebraic hyperbolic spaces are CAT(-1):
Proposition 3.2.3. Any Riemannian manifold (finite- or infinite-dimensional)
with sectional curvature bounded above by −1 is a CAT(-1) space.
Proof. The finite-dimensional case is proven in [39, Theorem II.1A.6]. The
infinite-dimensional follows upon augmenting the finite-dimensional proof with the
2The comparison points x and y may not be uniquely determined if either x or y lies on two sides

of the triangle simultaneously.

3.3. GROMOV HYPERBOLIC METRIC SPACES

29

infinite-dimensional Cartan–Hadamard theorem [119, IX, Theorem 3.8] to guarantee surjectivity of the exponential map.



Since algebraic hyperbolic spaces have sectional curvature bounded between
−4 and −1 (e.g. [93, Corollary of Proposition 4]; see also [147, Lemmas 2.3, 2.7,
and 2.11]), the following corollary is immediate:

Corollary 3.2.4. Every algebraic hyperbolic space is a CAT(-1) space.
Remark 3.2.5. One can prove Corollary 3.2.4 without using the full strength
of Proposition 3.2.3. Indeed, any geodesic triangle in an algebraic hyperbolic space
is isometric to a geodesic triangle in H2F for some F ∈ {R, C, Q}. Since H2F is finitedimensional, thinness of its geodesic triangles follows from the finite-dimensional
version of Proposition 3.2.3.
Observation 3.2.6. R-trees are CAT(-1).
Proof. First of all, an argument similar to the proof of Observation 3.2.2
shows that R-trees are uniquely geodesic, justifying Figure 3.1.1. In particular,
if ∆(p, q, r) is a geodesic triangle in an R-tree and if C = C(p, q, r) then [p, q] =
[p, C]∪[q, C], [q, r] = [q, C]∪[r, C], and [r, p] = [r, C]∪[p, C]. It follows that any two
points x, y ∈ ∆ share a side in common, without loss of generality say x, y ∈ [p, q].

Then

d(x, y) = d(p, q) − d(x, p) − d(y, q) = d(p, q) − d(x, p) − d(y, q) ≤ d(x, y).

In a sense R-trees are the “most negatively curved spaces”; although we did
not define the notion of a CAT(κ) space, R-trees are CAT(κ) for every κ ∈ R.
3.3. Gromov hyperbolic metric spaces
We now come to the theory of Gromov hyperbolic metric spaces. In a sense,
Gromov hyperbolic metric spaces are those which are “approximately R-trees”. A
good reference for this section is [172].
For any three numbers dpq , dqr , drp ≥ 0 satisfying the triangle inequality, there

exists an R-tree X and three points p, q, r ∈ X such that d(p, q) = dpq , etc. Thus
in some sense looking at triples “does not tell you” that you are looking at an
R-tree. Now let us look at quadruples. A quadruple (p, q, r, s) in an R-tree X looks
something like Figure 3.3.1. Of course, the points p, q, r, s ∈ X could be arranged
in any order. However, let us consider them the way that they are arranged in
Figure 3.3.1 and note that

30

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES
r

q

p

s

Figure 3.3.1. A quadruple of points in an R-tree

(3.3.1)

C(p, q, r) = C(p, q, s).

In order to write this equality in terms of distances, we need some way of measuring
the distance from the vertex of a geodesic triangle to its center.
Observation 3.3.1. If ∆(p, q, r) is a geodesic triangle in an R-tree then the
distance from the vertex p to the center C(p, q, r), i.e. d(p, C(p, q, r)), is equal to
(3.3.2)

hq|rip :=

1
[d(p, q) + d(p, r) − d(q, r)].
2

The expression hq|rip is called the Gromov product of q and r with respect to
p, and it makes sense in any metric space. It can be thought of as measuring the
“defect in the triangle inequality”; indeed, the triangle inequality is exactly what
assures that hq|rip ≥ 0 for all p, q, r ∈ X.
Now (3.3.1) implies that

hq|rip = hq|sip ≤ hr|sip .
(The last inequality does not follow from (3.3.1) but it may be seen from Figure
3.3.1.) However, since the arrangement of points was arbitrary we do not know
which two Gromov products will be equal and which one will be larger. An inequality which captures all possibilities is
(3.3.3)

hq|rip ≥ min(hq|sip , hr|sip ).

Now, as mentioned before, we will define hyperbolic metric spaces as those which
are “approximately R-trees”. Thus they will satisfy (3.3.3) with an asymptotic.
Definition 3.3.2. A metric space X is called hyperbolic (or Gromov hyperbolic)
if for every four points x, y, z, w ∈ X we have
(3.3.4)

hx|ziw &+ min(hx|yiw , hy|ziw ),

We refer to (3.3.4) as Gromov’s inequality.

3.3. GROMOV HYPERBOLIC METRIC SPACES

31

From the above discussion, every R-tree is Gromov hyperbolic with an implied
constant of 0 in (3.3.4). (This can also be deduced from Proposition 3.3.4 below.)
Note that many authors require X to be a geodesic metric space in order to
be hyperbolic; we do not. If X is a geodesic metric space, then the condition
of hyperbolicity can be reformulated in several different ways, including the thin
triangles condition; for details, see [39, § III.H.1] or Section 4.3 below.
It will be convenient for us to make a list of several identities satisfied by the
Gromov product. For each z ∈ X, let B z denote the Busemann function
(3.3.5)

B z (x, y) := d(z, x) − d(z, y).

Proposition 3.3.3. The Gromov product and Busemann function satisfy the
following identities and inequalities:
(a)

hx|yiz = hy|xiz

(b)

d(y, z) = hy|xiz + hz|xiy

(c)

0 ≤ hx|yiz ≤ min(d(x, z), d(y, z))

(d)

hx|yiz ≤ hx|yiw + d(z, w)

(e)

hx|yiw ≤ hx|ziw + d(y, z)
| B x (z, w)| ≤ d(z, w)

(f)

1
hx|yiz = hx|yiw + [B x (z, w) + B y (z, w)]
2
1
hx|yiz = [d(x, z) + B y (z, x)]
2
B x (y, z) = hz|xiy − hy|xiz

(g)
(h)
(j)

hx|yiz = hx|yiw + d(z, w) − hx|ziw − hy|ziw

(k)

1
hx|yiw = hx|ziw + [B w (y, z) − B x (y, z)]
2

(l)

The proof is a straightforward computation. We remark that (a)-(e) may be
found in [172, Lemma 2.8].
3.3.1. Examples of Gromov hyperbolic metric spaces.
Proposition 3.3.4 (Proven in Section 3.5). Every CAT(-1) space (in particular
every algebraic hyperbolic space) is Gromov hyperbolic. In fact, if X is a CAT(-1)
space then for every four points x, y, z, w ∈ X we have
(3.3.6)

e−hx|ziw ≤ e−hx|yiw + e−hy|ziw ,

and so X satisfies (3.3.4) with an implied constant of log(2).

32

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES
y
hz|xiy

z

hy|xiz

x

Figure 3.3.2. An illustration of (b) of Proposition 3.3.3 in an R-tree.
Remark 3.3.5. The first assertion of Proposition 3.3.4, namely, that CAT(-1)
spaces are Gromov hyperbolic, is [39, Proposition III.H.1.2]. The inequality (3.3.6)
in the case where x, y, z ∈ ∂X and w ∈ X can be found in [33, Théorème 2.5.1].
Definition 3.3.6. A space X satisfying the conclusion of Proposition 3.3.4 is
said to be strongly hyperbolic.
Note that
R-tree ⇒ CAT(-1) ⇒ Strongly hyperbolic ⇒ Hyperbolic.
A large class of examples of hyperbolic metric spaces which are not CAT(-1) is
furnished by the Cayley graphs of finitely presented groups. Indeed, we have the
following:
Theorem 3.3.7 ([86, p.78], [139]; see also [49]). Fix k ≥ 2 and an alpha-

±1
±1
bet A = {a±1
1 , a2 , · · · , ak }. Fix i ∈ N and a sequence of positive integers
(n1 , · · · , ni ). Let N = N (k, i, n1 , · · · , ni ) be the number of group presentations

G = ha1 , · · · , ak |r1 , · · · , ri i such that r1 , · · · , ri are reduced words in the alphabet A

such that the length of rj is nj for j = 1, 2, · · · , i. If Nh is the number of groups in
this collection whose Cayley graphs are hyperbolic and if n = min(n1 , · · · , ni ) then
limn→∞ Nh /N = 1.

This theorem says that in some sense, “almost every” finitely presented group
is hyperbolic.
If one has a hyperbolic metric space X, there are two ways to get another
hyperbolic metric space from X, one trivial and one nontrivial.
Observation 3.3.8. Any subspace of a hyperbolic metric space is hyperbolic.
Any subspace of a strongly hyperbolic metric space is strongly hyperbolic.
To describe the other method we need to define the notion of a quasi-isometric
embedding.

3.4. THE BOUNDARY OF A HYPERBOLIC METRIC SPACE

33

Definition 3.3.9. Let (X1 , d1 ) and (X2 , d2 ) be metric spaces. A map Φ :
X1 → X2 is a quasi-isometric embedding if for every x, y ∈ X1
d2 (Φ(x), Φ(y)) ≍+,× d1 (x, y).
A quasi-isometric embedding Φ is called a quasi-isometry if its image Φ(X1 ) is
cobounded in X2 , that is, if there exists R > 0 such that Φ(X1 ) is R-dense in X2 ,
meaning that minx∈X2 d(x, Φ(X1 )) ≤ R. In this case, the spaces X1 and X2 are
said to be quasi-isometric.
Theorem 3.3.10 ([39, Theorem III.H.1.9]). Any geodesic metric space which
can be quasi-isometrically embedded into a geodesic hyperbolic metric space is also
a hyperbolic metric space.
Remark 3.3.11. Theorem 3.3.10 is not true if the hypothesis of geodesicity is
dropped. For example, R is quasi-isometric to [0, ∞) × {0} ∪ {0} × [0, ∞) ⊆ R2 , but
the former is hyperbolic and the latter is not.

There are many more examples of hyperbolic metric spaces which we will not
discuss; cf. the list in §1.1.2.
3.4. The boundary of a hyperbolic metric space
In this section we define the Gromov boundary of a hyperbolic metric space
X. The construction will depend on a distinguished point o ∈ X, but the resulting
space will be independent of which point is chosen. If X is an R-tree, then the
boundary of X will turn out to be the set of infinite branches through X, i.e. the
set of all isometric embeddings π : [0, ∞) → X sending 0 to o, where o ∈ X is a

distinguished fixed point. If X is an algebraic hyperbolic space, then the boundary
of X will turn out to be isomorphic to the space ∂X defined in Chapter 2.
To motivate the definition of the boundary, suppose that X is an R-tree. An
infinite branch through X can be approximated by finite branches which agree
on longer and longer segments. Suppose that ([o, xn ])∞
1 is a sequence of geodesic
segments. For each n, m ∈ N, the length of the intersection of [o, xn ] and [o, xm ] is

equal to d(o, C(o, xn , xm )), which in turn is equal to hxn |xm io . Thus, the sequence
([o, xn ])∞
1 converges to an infinite geodesic if and only if
(3.4.1)

hxn |xm io −−→ ∞.
n,m

(Cf. Figure 3.4.1.) The formula (3.4.1) is reminiscent of the definition of a Cauchy
sequence. This intuition will be made explicit in Section 3.6, where we will introduce
a metametric on X with the property that a sequence in X satisfies (3.4.1) if and
only if it is Cauchy with respect to this metametric.

34

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES
x5
x3

x1

...

x2

o

Figure 3.4.1. A Gromov sequence in an R-tree
Definition 3.4.1. A sequence (xn )∞
1 in X for which (3.4.1) holds is called a
∞
Gromov sequence. Two Gromov sequences (xn )∞
1 and (yn )1 are called equivalent
if

hxn |yn io −
→ ∞,
n

or equivalently if
hxn |ym io −−→ ∞.
n,m

∞
In this case, we write (xn )∞
1 ∼ (yn )1 . It is readily verified using Gromov’s inequal-

ity that ∼ is an equivalence relation on the set of Gromov sequences in X. We will
∞
denote the class of sequences equivalent to a given sequence (xn )∞
1 by [(xn )1 ].

Definition 3.4.2. The Gromov boundary of X is the set of Gromov sequences
modulo equivalence. It is denoted ∂X. The Gromov closure or bordification of X
is the disjoint union bord X := X ∪ ∂X.
Remark 3.4.3. If X is an algebraic hyperbolic space, then this notation causes
some ambiguity, since it is not clear whether ∂X represents the Gromov boundary
of X, or rather the topological boundary of X as in Chapter 2. This ambiguity
will be resolved in §3.5.1 below when it is shown that the two bordifications are
isomorphic.
Remark 3.4.4. In the literature, the ideal boundary of a hyperbolic metric
space is often taken to be the set of equivalence classes of geodesic rays under
asymptotic equivalence, rather than the set of equivalence classes of Gromov sequences (e.g. [39, p.427]). If X is proper and geodesic, then these two notions are
equivalent [39, Lemma III.H.3.13], but in general they may be different.
Remark 3.4.5. By (d) of Proposition 3.3.3, the concepts of Gromov sequence
and equivalence do not depend on the basepoint o. In particular, the Gromov
boundary ∂X is independent of o.

3.4. THE BOUNDARY OF A HYPERBOLIC METRIC SPACE

35

3.4.1. Extending the Gromov product to the boundary. We now wish
to extend the Gromov product and Busemann function to the boundary “by continuity”. Fix ξ, η ∈ ∂X and z ∈ X. Ideally, we would like to define hξ|ηiz to
be

lim hxn |ym iz ,

(3.4.2)

n,m→∞

∞
where (xn )∞
1 ∈ ξ and (ym )1 ∈ η. (The definition would then have to be shown

independent of which sequences were chosen.) The naive definition (3.4.2) does not
work, because the limit (3.4.2) does not necessarily exist:
Example 3.4.6. Let
X = {x ∈ R2 : x2 ∈ [0, 1]}

be interpreted as a subspace of R2 with the L1 metric. Then X is a hyperbolic
metric space, since it contains the cobounded hyperbolic metric space R × {0}. Its
Gromov boundary consists of two points −∞ and +∞, which are the limits of x

as x1 approaches −∞ or +∞, respectively. Let y = (0, 1) and z = (1, 0). Then for
all x ∈ X, hx|yiz = x2 . In particular, we can find a sequence xn → +∞ such that

limn→∞ hxn |yiz does not exist.

Fortunately, the limit (3.4.2) “exists up to a constant”:
∞
Lemma 3.4.7. Let (xn )∞
1 and (ym )1 be Gromov sequences, and fix y, z ∈ X.

Then

lim inf hxn |ym iz ≍+ lim suphxn |ym iz

(3.4.3)

n,m→∞

n,m→∞

lim inf hxn |yiz ≍+ lim suphxn |yiz ,

(3.4.4)

n→∞

n→∞

with equality if X is strongly hyperbolic.
Note that except for the statement about strongly hyperbolic spaces, this
lemma is simply [172, Lemma 5.6].
Proof of Lemma 3.4.7. Fix n1 , n2 , m1 , m2 ∈ N. By Gromov’s inequality
hxn1 |ym1 iz &+ min(hxn2 |ym2 iz , hxn1 |xn2 iz , hym1 |ym2 iz ).
Taking the liminf over n1 , m1 and the limsup over n2 , m2 gives
lim inf hxn |ym iz


&+ min lim suphxn |ym iz , lim inf hxn1 |xn2 iz , lim inf hym1 |ym2 iz

n,m→∞

n,m→∞

= lim suphxn |ym iz ,
n,m→∞

m1 ,m2 →∞

n1 ,n2 →∞

(since

(xn )∞
1

and (ym )∞
1 are Gromov)

36

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

demonstrating (3.4.3). On the other hand, suppose that X is strongly hyperbolic.
Then by (3.3.6) we have




exp −hxn1 |ym1 iz ≤ exp −hxn2 |ym2 iz +exp −hxn1 |xn2 iz +exp −hym1 |ym2 iz ;

taking the limsup over n1 , m1 and the liminf over n2 , m2 gives


exp − lim inf hxn |ym iz ≤ exp − lim suphxn |ym iz +
n,m→∞

n,m→∞


+ exp − lim inf hxn1 |xn2 iz +
n1 ,n2 →∞

+ exp − lim inf hym1 |ym2 iz
m1 ,m2 →∞

= exp − lim suphxn |ym iz ,
n,m→∞

∞
(since (xn )∞
1 and (ym )1 are Gromov)

demonstrating equality in (3.4.3). The proof of (3.4.4) is similar and will be omitted.

Remark 3.4.8. Many of the statements in this monograph concerning strongly
hyperbolic metric spaces are in fact valid for all hyperbolic metric spaces satisfying
the conclusion of Lemma 3.4.7.
Now that we know that it does not matter too much whether we replace the
limit in (3.4.2) by a liminf or a limsup, we make the following definition without
fear:
Definition 3.4.9. For ξ, η ∈ ∂X and y, z ∈ X, let


∞
(3.4.5)
hξ|ηiz := inf lim inf hxn |ym iz : (xn )∞
∈
ξ,
(y
)
∈
η
m 1
1
n,m→∞
n
o
(3.4.6)
hξ|yiz := hy|ξiz := inf lim inf hxn |yiz : (xn )∞
1 ∈ξ
n→∞

(3.4.7)

B ξ (y, z) = hz|ξiy − hy|ξiz .

As a corollary of Lemma 3.4.7, we have the following:
∞
Lemma 3.4.10. Fix ξ, η ∈ ∂X and y, z ∈ X. For all (xn )∞
1 ∈ ξ and (ym )1 ∈ η

we have
(3.4.8)

hxn |ym iz −−−−→ hξ|ηiz

(3.4.9)

hxn |yiz −−−−→ hξ|yiz

(3.4.10)

B xn (y, z) −−−−→ B ξ (y, z),

n,m,+
n,+

n,+

(cf. Convention 2), with exact limits if X is strongly hyperbolic.

3.4. THE BOUNDARY OF A HYPERBOLIC METRIC SPACE

37

Note that except for the statement about strongly hyperbolic spaces, this
lemma is simply [172, Lemma 5.11].
(i)

(i)

∞
Proof of Lemma 3.4.10. Say we are given (xn )∞
1 ∈ ξ and (ym )1 ∈ η for
each i = 1, 2. Let

x(1)
n even
n/2
,
xn =
x(2)
n odd
(n+1)/2

and define ym similarly. It may be verified using Gromov’s inequality that (xn )∞
1 ∈
ξ and (ym )∞
1 ∈ η. Applying Lemma 3.4.7, we have
2

2

2

2

(i) (j)
(j)
min min lim inf hx(i)
n |ym iz ≍+ max max lim suphxn |ym iz .
i=1 j=1 n,m→∞

i=1 j=1 n,m→∞

In particular,
(1)
(1) (1)
lim inf hx(1)
n |yn iz .+ lim suphxn |yn iz

n,m→∞

n,m→∞

(2)
.+ lim inf hx(2)
n |yn iz
n,m→∞

(2)
(1) (1)
.+ lim suphx(2)
n |yn iz .+ lim inf hxn |yn iz .
n,m→∞

n,m→∞

(2)

(2)

∞
Taking the infimum over all (xn )∞
1 ∈ ξ and (ym )1 ∈ η gives (3.4.8). A similar
argument gives (3.4.9). Finally, (3.4.10) follows from (3.4.9), (3.4.7), and (j) of

Proposition 3.3.3.
If X is strongly hyperbolic, then all error terms are equal to zero, demonstrating
that the limits converge exactly.



Remark 3.4.11. In the sequel, the statement that “if X is strongly hyperbolic,
then all error terms are zero” will typically be omitted from our proofs.
A simple but useful consequence of Lemma 3.4.10 is the following:
Corollary 3.4.12. The formulas of Proposition 3.3.3 together with Gromov’s
inequality hold for points on the boundary as well, if the equations and inequalities
there are replaced by additive asymptotics. If X is strongly hyperbolic, then we may
keep the original formulas without adding an error term.
Proof. For each identity, choose a Gromov sequence representing each element of the boundary which appears in the formula. Replace each occurrence of
this element in the formula by the general term of the chosen sequence. This yields
a sequence of formulas, each of which is known to be true. Take a subsequence
on which each term in these formulas converges. Taking the limit along this subsequence again yields a true formula, and by Lemma 3.4.10 we may replace each
limit term by the term which it stood for, with only bounded error in doing so,

38

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

and no error if X is strongly hyperbolic. Thus the formula holds as an additive
asymptotic, and holds exactly if X is strongly hyperbolic.



Remark 3.4.13. In fact, (a), (c), (d), and (e) of Proposition 3.3.3 hold in
bord X in the usual sense, i.e. as exact formulas without additive constants.
Proof. These are the identities where there is at most one Gromov product
on each side of the formula. For each element of the boundary, we may simply
replace each occurence of that element with the general term of an arbitrary Gromov
sequence, take the liminf, and then take the infimum over all Gromov sequences. 
Observation 3.4.14. hx|yiz = ∞ if and only if x = y ∈ ∂X.
Proof. This follows directly from (3.4.5) and (3.4.6).



3.4.2. A topology on bord X. One can endow the bordification bord X =
X ∪ ∂X with a topological structure T as follows: Given S ⊆ bord X, write S ∈ T
(i.e. call S open) if
(I) S ∩ X is open, and

(II) For each ξ ∈ S ∩ ∂X there exists t ≥ 0 such that Nt (ξ) ⊆ S, where
Nt (ξ) := Nt,o (ξ) := {y ∈ bord X : hy|ξio > t}
Remark 3.4.15. The topology T may equivalently be defined to be the unique
topology on bord X satisfying:
(I) T ↿ X is compatible with the metric d, and
(II) For each ξ ∈ ∂X, the collection
{Nt (ξ) : t ≥ 0}

(3.4.11)

is a neighborhood base for T at ξ.
Remark 3.4.16. It follows from Lemma 3.4.23 below that the sets Nt (ξ) are
open in the topology T .
Remark 3.4.17. By (d) of Proposition 3.3.3 (cf. Remark 3.4.13), we have
Nt,x (ξ) ⊇ Nt+d(x,y),y (ξ) for all x, y ∈ X, ξ ∈ ∂X, and t ≥ 0. Thus the topology T

is independent of the basepoint o.

The topology T is quite nice. In fact, we have the following:
Proposition 3.4.18. The topological space (bord X, T ) is completely metrizable. If X is proper and geodesic, then bord X (and thus also ∂X) is compact. If
X is separable, then bord X (and thus also ∂X) is separable.
Remark 3.4.19. If X is proper and geodesic, then Proposition 3.4.18 is [39,
Exercise III.H.3.18(4)].

3.4. THE BOUNDARY OF A HYPERBOLIC METRIC SPACE

39

Proof of Proposition 3.4.18. We delay the proof of the complete metrizability of bord X until Section 3.6, where we will introduce a class of compatible
complete metrics on bord X which are important from a geometric point of view,
the so-called visual metrics.
Since X is dense in bord X, the separability of X implies the separability of
bord X. Moreover, since bord X is metrizable (as we will show in Section 3.6), the
separability of bord X implies the separability of ∂X.
Finally, assume that X is proper and geodesic; we claim that bord X is compact.
∞
Let (xn )∞
1 be a sequence in X. If (xn )1 contains a bounded subsequence, then
since X is proper it contains a convergent subsequence. Thus we assume that (xn )∞
1
contains no bounded subsequence, i.e. kxn k → ∞.
For each n ∈ N and t ≥ 0 let

xn,t = [o, xn ]t∧kxn k , 3
where [o, xn ] is any geodesic connecting o and xn . Since X is proper, there exists a
∞
sequence (nk )∞
1 such that for each t ≥ 0, the sequence (xnk ,t )1 is convergent, say

xnk ,t −
→ xt .
k

It is readily verified that the map t 7→ xt is an isometric embedding from [0, ∞) to

X. Thus there exists a point ξ ∈ ∂X such that xt → ξ. We claim that xnk → ξ.
Indeed, for each t ≥ 0,
lim sup D(xnk , xnk ,t ) ≍× lim sup D(xnk ,t , xt ) ≍× lim sup D(xt , ξ) ≍× b−t ,
k→∞

k→∞

k→∞

and so the triangle inequality gives
lim sup D(xnk , ξ) .× b−t .
k→∞

Letting t → ∞ shows that xnk → ξ.



Observation 3.4.20. A sequence (xn )∞
1 in bord X converges to a point ξ ∈ ∂X
if and only if
(3.4.12)

hxn |ξio −
→ ∞.
n

Observation 3.4.21. A sequence (xn )∞
1 in X converges to a point ξ ∈ ∂X if

∞
and only if (xn )∞
1 is a Gromov sequence and (xn )1 ∈ ξ.

We now investigate the continuity properties of the Gromov product and Busemann function.
Lemma 3.4.22 (Near-continuity of the Gromov product and Busemann function). The maps (x, y, z) 7→ hx|yiz and (x, z, w) 7→ B x (z, w) are nearly continuous
3Here and from now on A ∧ B = min(A, B) and A ∨ B = max(A, B).

40

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

∞
in the following sense: Suppose that (xn )∞
1 and (yn )1 are sequences in bord X

which converge to points xn → x ∈ bord X and yn → y ∈ bord X. Suppose that
∞
(zn )∞
1 and (wn )1 are sequences in X which converge to points zn → z ∈ X and

wn → w ∈ X. Then
(3.4.13)

hxn |yn izn −−→ hx|yiz

(3.4.14)

B xn (zn , wn ) −−→ B x (z, w),

n,+

n,+

with −
→ if X is strongly hyperbolic.
n

Proof. In the proof of (3.4.13), there are three cases:
Case 1: x, y ∈ X. In this case, (3.4.13) follows directly from (d) and (e) of Proposition 3.3.3.
Case 2: x, y ∈ ∂X. In this case, for each n ∈ N, choose x
bn ∈ X such that either
(1) x
bn = xn (if xn ∈ X), or
(2) hb
xn |xn iz ≥ n (if xn ∈ ∂X).

(3.4.15)

Choose ybn similarly. Clearly, x
bn → x and ybn → y. By Observation 3.4.21,
∞
∞
yn )1 ∈ y. Thus by Lemma 3.4.10,
(b
xn )1 ∈ x and (b
hb
xn |b
yn iz −−→ hx|yiz .
n,+

Now by Gromov’s inequality and (e) of Proposition 3.3.3, either
(1)

hb
xn |b
yn iz ≍+ hxn |yn izn or

(2)

hb
xn |b
yn iz &+ n,
with which asymptotic is true depending on n. But for n sufficiently large,
(3.4.15) ensures that the (2) fails, so (1) holds.

Case 3: x ∈ X, y ∈ ∂X, or vice-versa. In this case, a straightforward combination
of the above arguments demonstrates (3.4.13).
Finally, note that (3.4.14) is an immediate consequence of (3.4.13), (3.4.7), and (j)
of Proposition 3.3.3.



Although Lemma 3.4.22 is generally sufficient for applications, we include the
following lemma which reassures us that the Gromov product does behave somewhat regularly even on an “exact” level.
Lemma 3.4.23. The function (x, y, z) 7→ hx|yiz is lower semicontinuous on
bord X × bord X × X.
Proof. Since bord X is metrizable, it is enough to show that if xn → x,

yn → y, and zn → z, then

lim inf hxn |yn izn ≥ hx|yiz .
n→∞

3.5. THE GROMOV PRODUCT IN ALGEBRAIC HYPERBOLIC SPACES

41

Now fix ε > 0.
Claim 3.4.24. For each n ∈ N, there exist points x
bn , ybn ∈ X satisfying:
hb
xn |b
yn izn ≤ hxn |yn izn + ε,

(3.4.16)

hb
xn |xn io ≥ n, or x
bn = xn ∈ X,

(3.4.17)

hb
yn |yn io ≥ n, or ybn = yn ∈ X.

(3.4.18)

Proof. Suppose first that xn , yn ∈ ∂X. By the definition of hxn |yn izn , there

∞
exist (xn,k )∞
1 ∈ xn and (yn,ℓ )1 ∈ yn such that

lim inf hxn,k |yn,ℓ izn ≤ hxn |yn izn + ε/2.
k,ℓ→∞

It follows that there exist arbitrarily large4 pairs (k, ℓ) ∈ N2 such that the points

x
bn := xn,k and ybn := yn,ℓ satisfy (3.4.16). Since (3.4.17) and (3.4.18) are satisfied
for all sufficiently large (k, ℓ) ∈ N2 , this completes the proof. Finally, if either

xn ∈ X, yn ∈ X, or both, a straightforward adaptation of the above argument
yields the claim.
⊳
The equations (3.4.17) and (3.4.18), together with Gromov’s inequality, imply
that x
bn → x and ybn → y. Now suppose that x, y ∈ ∂X. Then by Observation
3.4.21, (b
xn )∞
yn )∞
1 ∈ x and (b
1 ∈ y. So by the definition of hx|yiz , we have
hx|yiz ≤ lim inf hb
xn |b
yn iz
n→∞

= lim inf hb
xn |b
yn izn
n→∞

≤ lim inf hxn |yn izn + ε.
n→∞

(by the definition of hx|yiz )

(by (d) of Proposition 3.3.3)
(by (3.4.16))

Letting ε tend to zero completes the proof. A similar argument applies to the case
where x ∈ X, y ∈ X, or both.



Lemma 3.4.25. If g is an isometry of X, then it extends in a unique way to a
continuous map ge : bord X → bord X.

Proof. This follows more or less directly from Remarks 3.4.5 and 3.4.17; de-

tails are left to the reader.



In the sequel we will omit the tilde from the extended map ge.
3.5. The Gromov product in algebraic hyperbolic spaces
In this section we analyze the Gromov product in an algebraic hyperbolic space
X. We prove Proposition 3.3.4 which states that CAT(-1) spaces are strongly
4Here, of course, “arbitrarily large” means that min(k, ℓ) can be made arbitrarily large.

42

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

x

y

θ
0

Figure 3.5.1. If F = R and x, y ∈ ∂B, then e−hx|yi0 = 12 ky−xk =
sin(θ/2), where θ denotes the angle ∡0 (x, y) drawn in the figure.
hyperbolic, and then we show that the Gromov boundary of X is isomorphic to its
topological boundary, justifying Remark 3.4.3.
In what follows, we will switch between the hyperboloid model H = Hα
F and
α
the ball model B = BF according to convenience. In the following lemma, ∂B and
bord B denote the topological boundary and closure of B as defined in Chapter 2,
not the Gromov boundary and closure as defined above.
Lemma 3.5.1. The Gromov product (x, y, z) 7→ hx|yiz : B × B × B → [0, ∞)

extends uniquely to a continuous function (x, y, z) 7→ hx|yiz : bord B×bord B×B →
[0, ∞]. Moreover, the extension satisfies the following:
(i) hx|yiz = ∞ if and only if x = y ∈ ∂B.

(ii) For all x, y ∈ bord B,
(3.5.1)

1
e−hx|yi0 ≥ √ ky − xk.
8

If F = R and x, y ∈ ∂B, then

1
ky − xk.
2
Proof. We begin by making some computations in the hyperboloid model H.
For [x], [y] ∈ bord H and [z] ∈ H, let

(3.5.2)

e−hx|yi0 =

α[z] ([x], [y]) =

|Q(z)| · |BQ (x, y)|
∈ [0, ∞).
|BQ (x, z)| · |BQ (y, z)|

By (2.2.2), for [x], [y], [z] ∈ H we have
(3.5.3)

α[z] ([x], [y]) =

cosh dH ([x], [y])
·
cosh dH ([x], [z]) cosh dH ([y], [z])

3.5. THE GROMOV PRODUCT IN ALGEBRAIC HYPERBOLIC SPACES

43

3

Let D = {(A, B, C) ∈ [0, ∞) : cosh(A) cosh(B)C ≥ 1}, and define F : D → [0, ∞)

by



exp cosh−1 (cosh(A) cosh(B)C)
F (A, B, C) =
·
eA eB
Then by (3.5.3), we have

e−2h[x]|[y]i[z] = F dH ([z], [x]), dH ([z], [y]), α[z] ([x], [y])

for all [x], [y], [z] ∈ H. Now since limt→∞
C>0

et
cosh(t)

= 2, we have for all A ≥ 0 and

2 (cosh(A) cosh(B)C)
eA eB
cosh(A)
=
C
eA

lim F (A, B, C) = lim

B→∞

B→∞

and

cosh(A)
C = C/2.
A→∞
eA
b be the closure of D relative to [0, ∞]2 × [0, ∞), i.e.
Let D


b = D ∪ [0, ∞]2 × [0, ∞) \ [0, ∞)3 .
D
lim

If we let

Fb (A, B, C) :=




F (A, B, C)




 cosh(A) C
eA
cosh(B)


C

 eB



C/2

A, B < ∞
A<B=∞
B<A=∞
A=B=∞

b → [0, ∞) is a continuous function.5 Thus, letting
then Fb : D


1
h[x]|[y]i[z] := − log Fb dH ([z], [x]), dH ([z], [y]), α[z] ([x], [y])
2
defines a continuous extension of the Gromov product to bord H × bord H × H.
We now prove (i)-(ii):
(i) Using the inequality et /2 ≤ cosh(t) ≤ et , it is easily verified that
Fb (A, B, C) ≥ C/4

(3.5.4)

b In particular, if Fb (A, B, C) = 0 then C = 0. Thus if
for all (A, B, C) ∈ D.

h[x]|[y]i[z] = ∞ then α[z] ([x], [y]) = 0; since [z] ∈ H we have BQ (x, y) = 0,
and by Observation 2.3.15 we have [x] = [y] ∈ ∂H. Conversely, if [x] =
[y] ∈ ∂H, then dH ([z], [x]) = dH ([z], [y]) = ∞ and α[z] ([x], [y]) = 0, so
h[x]|[y]i[z] = − 1 log Fb(∞, ∞, 0) = ∞.
2

5Technically, the calculations above do not prove the continuity of F
b; however, this continuity is

easily verified using standard methods.

44

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

(ii) Recall that in B, o = [(1, 0)]. For x, y ∈ B,
αo (eB,H (x), eB,H (y)) =

|Q(1, 0)| · |BQ ((1, x), (1, y))|
|BQ ((1, 0), (1, x))| · |BQ ((1, 0), (1, y))|

= |1 − BE (x, y)|

≥ 1 − Re BE (x, y)

1
[kxk2 + kyk2 ] − Re BE (x, y)
2
1
= ky − xk2 .
2

≥

(with equality if F = R)
(with equality if x, y ∈ ∂B)

Combining with (3.5.4) gives
1
1
αo (eB,H (x), eB,H (y)) ≥ ky − xk2 .
4
8
If F = R and x, y ∈ ∂B, then
e−2hx|yi0 ≥

e−2hx|yi0 =

1
1
αo (eB,H (x), eB,H (y)) = ky − xk2 .
2
4


We now prove Proposition 3.3.4, beginning with the following lemma:

Lemma 3.5.2. If F = R then B is strongly Gromov hyperbolic.

Proof. By the transitivity of the isometry group (Observation 2.3.2), it suffices to check (3.3.6) for the special case w = o. So let us fix x, y, z ∈ B, and by

contradiction suppose that

e−hx|zio > e−hx|yio + e−hy|zio ,
or equivalently that
1 > ehx|zio −hx|yio + ehx|zio −hy|zio .
Clearly, the above inequality implies that x 6= z and y 6= o. Now let γ1 and γ2

be the unique bi-infinite geodesics extending the geodesic segments [x, z] and [o, y],
respectively. Let x∞ , z∞ ∈ ∂B be the appropriate endpoints of γ1 , and let y∞ be

the endpoint of γ2 which is closer to y than to o. (See Figure 3.5.2.) For each
t ∈ [0, ∞), let
xt = [x, x∞ ]t ∈ γ1 ,

and let yt ∈ γ2 , zt ∈ γ1 be defined similarly.

3.5. THE GROMOV PRODUCT IN ALGEBRAIC HYPERBOLIC SPACES

45

x∞

y∞

xt
x
yt
y

o

z
zt

z∞

Figure 3.5.2. If Gromov’s inequality fails for the quadruple
x, y, z, o, then it also fails for the quadruple x∞ , y∞ , z∞ , o.
We observe that

1 ∂
∂
dB (o, zt ) + dB (xt , yt ) − dB (xt , zt ) − dB (o, yt )
[hxt |zt io − hxt |yt io ] =
∂t
2 ∂t

1 ∂
=
dB (o, zt ) + dB (xt , yt ) − 2t − t
2 ∂t

1 ∂
t + 2t − 2t − t = 0,
≤
2 ∂t

i.e. the expression hxt |zt io − hxt |yt io is nonincreasing with respect to t. Taking the
limit as t approaches infinity, we have
hx∞ |z∞ io − hx∞ |y∞ io ≤ hx|zio − hx|yio
and a similar argument shows that
hx∞ |z∞ io − hy∞ |z∞ io ≤ hx|zio − hy|zio .
Thus
1 > ehx∞ |z∞ io −hx∞ |y∞ io + ehx∞ |z∞ io −hy∞ |z∞ io
or equivalently,
e−hx∞ |z∞ io > e−hx∞ |y∞ io + e−hy∞ |z∞ io .
But by (3.5.2), if we write x∞ = x, y∞ = y, and z∞ = z, then
1
1
1
kz − xk > ky − xk + kz − yk.
2
2
2
This is a contradiction.
We are now ready to prove



46

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

Proposition 3.3.4. Every CAT(-1) space is strongly hyperbolic.
Proof. Let X be a CAT(-1) space, and fix x, y, z, w ∈ X. By [39, Proposition
II.1.11], there exist x, y, z, w ∈ H2 such that
d(x, y) = d(x, y)

d(y, z) = d(y, z)

d(z, w) = d(z, w)

d(w, x) = d(w, x)

d(x, z) ≤ d(x, z)

d(y, w) ≤ d(y, w).

It follows that
e−hx|ziw ≤ e−hx|ziw ≤ e−hx|yiw + e−hy|ziw ≤ e−hx|yiw + e−hy|ziw .


3.5.1. The Gromov boundary of an algebraic hyperbolic space. Again
let X = H = Hα
F be an algebraic hyperbolic space. By Proposition 3.3.4, X is a
(strongly) hyperbolic metric space. (If F = R, we can use Lemma 3.5.2.) In
particular, X has a Gromov boundary, defined in Section 3.4. On the other hand,
X also has a topological boundary, defined in Chapter 2. For this subsection only,
we will write
∂G X = Gromov boundary of X,
∂T X = topological boundary of X.
We will now show that this distinction is in fact unnecessary.
Proposition 3.5.3. The identity map id : X → X extends uniquely to a

b : X ∪ ∂G X → X ∪ ∂T X. Thus the pairs (X, X ∪ ∂G X) and
homeomorphism id
(X, X ∪ ∂T X) are isomorphic in the sense of Section 2.4.
Proof of Proposition 3.5.3. By Observation 2.5.1 and Proposition 2.5.5,
it suffices to consider the case where X = B = Bα
F is the ball model. Fix ξ ∈ ∂G B.
∞
By definition, ξ = [(xn )∞
1 ] for some Gromov sequence (xn )1 . By (3.5.1), the

sequence (xn )∞
1 is Cauchy in the metric k · − · k. Thus xn → x for some x ∈ bord B;
is
a Gromov sequence, we have
since (xn )∞
1
hx|xi0 =

lim hxn |xm i0 = ∞,

n,m→∞

and thus x ∈ ∂T B by (i) of Lemma 3.5.1. Let
b
id(ξ)
= x.

3.5. THE GROMOV PRODUCT IN ALGEBRAIC HYPERBOLIC SPACES

47

b is well-defined, note that if (yn )∞ is another Gromov
To see that the map id
1
sequence equivalent to (xn )∞
1 , and if yn → y ∈ ∂T B, then
hx|yi0 = lim hxn |yn i0 = ∞,
n→∞

and so by (i) of Lemma 3.5.1 we have x = y.
b : ∂G B → ∂T B is a bijection. To demonstrate injectivity,
We next claim that id
b
b
we note that if id(ξ) = id(η) = x, then by (i) of Lemma 3.5.1
lim hxn |yn i0 = hx|xi0 = ∞,

n→∞

∞
where (xn )∞
1 and (yn )1 are Gromov sequences representing ξ and η, respectively.
∞
Thus (xn )∞
1 and (yn )1 are equivalent, and so ξ = η.
To demonstrate surjectivity, we observe that for x ∈ ∂T B, we have
∞ 

n−1
b
x
= x.
id
n
1

b is a homeomorphism, or in other words that
Finally, we must demonstrate that id
the topology defined in §3.4.2 is the usual topology on bord B (i.e. the topology
inherited from H). It suffices to demonstrate the following:

Claim 3.5.4. For any x ∈ ∂T B, the collection (3.4.11) (with ξ = x) is a

neighborhood base of x with respect to the usual topology.
Proof. By (3.5.1), we have
√
Nt (x) ⊆ B(x, 8e−t ).

On the other hand, the continuity of the Gromov product on bord B guarantees
that Nt (x) contains a neighborhood of x with respect to the usual topology.

⊳


In the sequel, the following will be useful:
Proposition 3.5.5. Let E = Eα be the half-space model of a real hyperbolic
space. For x, y ∈ E, we have
B ∞ (x, y) = − log(x1 /y1 ).
Proof. By (2.5.3) we have
eB∞ (x,y) = lim

z→∞

cosh dE (z, x)
exp dE (z, x)
= lim
exp dE (z, y) z→∞ cosh dE (z, y)
= lim

z→∞

1+
1+

kz−xk2
2x1 z1
kz−yk2
2y1 z1



= lim 
z→∞

kz−xk2
2x1 z1
kz−yk2
2y1 z1



 =

y1
·
x1


48

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES
∞

x
y
x1

y1

Figure 3.5.3. The value of the Busemann function B ∞ (x, y) depends on the heights of the points x and y.
3.6. Metrics and metametrics on bord X
3.6.1. General theory of metametrics.
Definition 3.6.1. Recall that a metric on a set Z is a map D : Z × Z → [0, ∞)

which satisfies:

(I) Reflexivity: D(x, x) = 0.
(II) Reverse reflexivity: D(x, y) = 0 ⇒ x = y.

(III) Symmetry: D(x, y) = D(y, x).
(IV) Triangle inequality: D(x, z) ≤ D(x, y) + D(y, z).

Now we can define a metametric on Z to be a map D : Z × Z → [0, ∞) which
satisfies (II), (III), and (IV), but not necessarily (I). This concept is not to be
confused with the more common notion of a pseudometric, which satisfies (I), (III),

and (IV), but not necessarily (II). The term “metametric” was introduced by J.
Väisälä in [172].
If D is a metametric, we define its domain of reflexivity to be the set Zrefl :=
{x ∈ Z : D(x, x) = 0}.6 Obviously, D restricted to its domain of reflexivity is a
metric.
As in metric spaces, a sequence (xn )∞
1 in a metametric space (Z, D) is called
Cauchy if D(xn , xm ) −−→ 0, and convergent if there exists x ∈ Z such that
n,m

D(xn , x) → 0. (However, see Remark 3.6.5 below.) The metametric space (Z, D) is
called complete if every Cauchy sequence is convergent. Using these definitions, the
standard proof of the Banach contraction principle immediately yields the following:
Theorem 3.6.2 (Banach contraction principle for metametric spaces). Let
(Z, D) be a complete metametric space. Fix 0 < λ < 1. If g : Z → Z satisfies
D(g(z), g(w)) ≤ λD(z, w) ∀z, w ∈ Z,
6In the terminology of [172, p.19], the domain of reflexivity is the set of “small points”.

3.6. METRICS AND METAMETRICS ON bord X

49

then there exists a unique point z ∈ Z so that g(z) = z. Moreover, for all w ∈ Z,

we have g n (w) → z with respect to the metametric D.

Observation 3.6.3. The fixed point coming z coming from Theorem 3.6.2
must lie in the domain of reflexivity Zrefl .
Proof.
D(z, z) = D(g(z), g(z)) ≤ λD(z, z),
and thus D(z, z) = 0.



Recall that a metric is said to be compatible with a topology if that topology
is equal to the topology induced by the metric. We now generalize this concept by
introducing the notion of compatibility between a topology and a metametric.
Definition 3.6.4. Let (Z, D) be a metametric space. A topology T on Z is
compatible with the metametric D if for every ξ ∈ Zrefl , the collection
(3.6.1)

{BD (ξ, r) := {y ∈ Z : D(ξ, y) < r} : r > 0}

forms a neighborhood base for T at ξ.
Note that unlike a metric, a metametric may have multiple topologies with
which it is compatible.7 The metametric is viewed as determining a neighborhood
base for points in the domain of reflexivity; neighborhood bases for other points
must arise from some other structure. In the case we are interested in, namely
the case where the underlying space for the metametric is the Gromov closure of
a hyperbolic metric space X, the topology on the complement of the domain of
reflexivity will come from the original metric d on X.
Remark 3.6.5. If (Z, D) is a metametric space with a compatible topology T ,
then there are two notions of what it means for a sequence (xn )∞
1 in Z to converge
to a point x ∈ Z: the sequence may converge with respect to the topology T , or

it may converge with respect to the metametric (i.e. D(xn , x) → 0). The relation
between these notions is as follows: xn → x with respect to the metametric D if
and only if both of the following hold: xn → x with respect to the topology T ,
and x ∈ Zrefl.

Remark 3.6.6. If a metametric D on a set Z is compatible with a topology
T , then the metric D ↿ Zrefl is compatible with the topology T ↿ Zrefl. However,
the converse does not necessarily hold.
For the remainder of this chapter, we fix a hyperbolic metric space X, and we
let T be the topology on bord X introduced in §3.4.2. We will consider various
metrics and metametrics on bord X which are compatible with the topology T .
7The topology considered in [172, p.19] is the finest topology compatible with a given metametric.

50

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

3.6.2. The visual metametric based at a point w ∈ X. The first meta-

metric that we will consider is designed to emulate the Euclidean or “spherical”
metric on the boundary of the ball model B. Recall from Lemma 3.5.1 that
(3.5.2)

1
ky − xk = e−hx|yi0 for all x, y ∈ ∂B.
2

The metric (x, y) 7→ 21 ky−xk can be thought of as “seen from 0”. The expression on
the right hand side makes sense if x, y ∈ bord B, and defines a metametric on bord B.

Moreover, the formula can be generalized to an arbitrary strongly hyperbolic metric
space:

Observation 3.6.7. If X is a strongly hyperbolic metric space, then for each
w ∈ X the map Dw : bord X × bord X → [0, ∞) defined by
(3.6.2)

Dw (x, y) := e−hx|yiw

is a complete metametric on bord X. This metametric is compatible with the
topology T ; moreover, its domain of reflexivity is ∂X.
Proof. Reverse reflexivity and the fact that (bord X)refl = ∂X follow directly
from Observation 3.4.14; symmetry follows from (a) of Proposition 3.3.3 together
with Corollary 3.4.12; the triangle inequality follows from the definition of strong
hyperbolicity together with Corollary 3.4.12.
To show that Dw is complete, suppose that (xn )∞
1 is a Cauchy sequence in
X. Applying (3.6.2), we see that hxn |xm iw −−→ ∞, i.e. (xn )∞
1 is a Gromov
n,m

sequence. Letting ξ = [(xn )∞
1 ], we have xn → ξ in the Db,w metametric. Thus
every Cauchy sequence in X converges in bord X. Since X is dense in bord X, a

standard approximation argument shows that bord X is complete.
Given ξ ∈ (bord X)refl = ∂X, the collection (3.6.1) is equal to the collection

(3.4.11), and is therefore a neighborhood base for T at ξ. Thus Dw is compatible
with T .


Next, we drop the assumption that X is strongly hyperbolic. Fix b > 1 and
w ∈ X, and consider the function
(3.6.3)

Db,w (x, y) = infn
(xi )0

n−1
X

b−hxi |xi+1 iw ,

i=0

where the infimum is taken over finite sequences (xi )n0 satisfying x0 = x and xn = y.
Proposition 3.6.8. If b > 1 is sufficiently close to 1, then for each w ∈ X,

the function Db,w defined by (3.6.3) is a complete metametric on bord X satisfying
the following inequality:
(3.6.4)

b−hx|yiw /4 ≤ Db,w (x, y) ≤ b−hx|yiw .

3.6. METRICS AND METAMETRICS ON bord X

51

This metametric is compatible with the topology T ; moreover, its domain of reflexivity is ∂X.
We will refer to Db,w as the “visual (meta)metric from the point w with respect
to the parameter b”.
Remark 3.6.9. The metric Db,w ↿ ∂X has been referred to in the literature as
the Bourdon metric.
Remark 3.6.10. The first part of Proposition 3.6.8 is [172, Propositions 5.16
and 5.31].
Proof of Proposition 3.6.8. Let δ ≥ 0 be the implied constant in Gro-

mov’s inequality, and fix 1 < b ≤ 21/δ . Then raising b−1 to the power of both sides
of Gromov’s inequality gives


b−hx|ziw ≤ 2 max b−hx|yiw , b−hy|ziw ,
i.e. the function

(x, y) 7→ b−hx|yiw
satisfies the “weak triangle inequality” of [154]. A straightforward adaptation of
the proof of [154, Theorem 1.2] demonstrates (3.6.4). Condition (II) of being a
metametric and the equality (bord X)refl = ∂X now follow from Observation 3.4.14.
Conditions (III) and (IV) of being a metametric are immediate from (3.6.3).
The argument for completeness is the same as in the proof of Observation 3.6.7.
Finally, given ξ ∈ (bord X)refl = ∂X, we observe that although the collections

(3.6.1) and (3.4.11) are no longer equal, (3.6.4) guarantees that the filters they
generate are equal, which is enough to show that Db,w is compatible with T . 
Remark 3.6.11. If X is strongly hyperbolic, then Proposition 3.6.8 holds for
all 1 < b ≤ e; moreover, the metametric De,w is equal to the metametric Dw defined
in Observation 3.6.7.

Remark 3.6.12. If (X, d) is an R-tree, then for all t > 0, (X, td) is also an
R-tree and is therefore strongly hyperbolic (by Observation 3.2.6 and Proposition
3.3.4). It follows that Proposition 3.6.8 holds for all b > 1.
For the remainder of this chapter, we fix b > 1 close enough to 1 so that
Proposition 3.6.8 holds.
3.6.3. The extended visual metric on bord X. Although the metametric
Db,w has the advantage of being directly linked to the Gromov product via (3.6.4),
it is sometimes desirable to put a metric on bord X, not just a metametric. We
show now that such a metric can be constructed which agrees with Db,w on ∂X.

52

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

In the following proposition, we use the convention that d(x, y) = ∞ if x, y ∈

bord X and either x ∈ ∂X or y ∈ ∂X.

Proposition 3.6.13. Fix w ∈ X, and for all x, y ∈ bord X let

Db,w (x, y) = min log(b)d(x, y), Db,w (x, y) .

Then D = Db,w is a complete metric on bord X which agrees with D = Db,w on
∂X and induces the topology T .
We call the metric D an extended visual metric.
As an immediate consequence we have the following result which was promised
in §3.4.2:
Corollary 3.6.14. The topological space (bord X, T ) is completely metrizable.
Proof of Proposition 3.6.13. Let us show that D is a metric. Conditions
(I)-(III) are obvious. To demonstrate the triangle inequality, fix x, y, z ∈ bord X.

(1) If D(x, y) = log(b)d(x, y) and D(y, z) = log(b)d(y, z), then D(x, z) ≤
log(b)d(x, z) ≤ log(b)d(x, y) + log(b)d(y, z) = D(x, y) + D(y, z). Similarly,
if D(x, y) = D(x, y) and D(y, z) = D(y, z), then D(x, z) ≤ D(x, z) ≤
D(x, y) + D(y, z) = D(x, y) + D(y, z).

(2) If D(x, y) = log(b)d(x, y) and D(y, z) = D(y, z), fix ε > 0, and let y =
y0 , y1 , . . . , yn = z be a sequence such that
n−1
X
i=0

b−hyi ,yi+1 iw ≤ D(y, z) + ε.

Let xi = yi for i ≥ 1 but let x0 = x. Then by (e) of Proposition 3.3.3 and

the inequality

b−t ≤ s log(b) + b−(t+s)

(s, t ≥ 0),

we have
b−hx|y1 iw ≤ log(b)d(x, y) + b−hy|y1 iw .
It follows that
D(x, z) ≤ D(x, z) ≤

n−1
X

b−hxi |xi+1 iw

i=0

= b−hx|y1 iw +

n−1
X

b−hyi |yi+1 iw

i=1

≤ log(b)d(x, y) + b−hy|y1 iw +

n−1
X

b−hyi |yi+1 iw

i=1

≤ log(b)d(x, y) + D(y, z) + ε = D(x, y) + D(y, z) + ε.

3.6. METRICS AND METAMETRICS ON bord X

53

Taking the limit as ε goes to zero finishes the proof.
(3) The third case is identical to the second.
If a sequence (xn )∞
1 is Cauchy with respect to D, then Ramsey’s theorem (for
example) guarantees that some subsequence is Cauchy with respect to either d or
D. This subsequence converges with respect to that metametric, and therefore also
with respect to D. It follows that the entire sequence converges, and therefore that
D is complete.
Finally, to show that D induces the topology T , suppose that U ⊆ ∂X is
open in T , and fix x ∈ U . If x ∈ X, then Bd (x, r) ⊆ U for some r > 0. On the

other hand, by the triangle inequality D(x, y) ≥ 12 D(x, x) > 0 for all y ∈ bord X.

Letting re = min(r, 21 D(x, x)), we have BD (x, re) ⊆ Bd (x, r) ⊆ U . If x ∈ ∂X, then
Nt (x) ⊆ U for some t ≥ 0; letting C be the implied constant of (3.6.4), we have

BD (x, e−t /C) = BD (x, e−t /C) ⊆ Nt (x) ⊆ U . Thus U is open in the topology
generated by the D metric. The converse direction is similar but simpler, and will
be omitted.



Remark 3.6.15. The proof of Proposition 3.6.13 actually shows more, namely
that
D(x, z) ≤ D(x, y) + D(y, z) ∀x, y, z ∈ bord X.

Since D(x, x) = b−kxk = inf y∈bord X D(x, y), plugging in x = z gives
b−kxk ≤ b−kyk + D(x, y) ∀x, y ∈ bord X.
Remark 3.6.16. Although the metric D is convenient since it induces the
correct topology on bord X, it is not a generalization of the Euclidean metric on
the closure of an algebraic hyperbolic space. Indeed, when X = B2 , then D is not
bi-Lipschitz equivalent to the Euclidean metric on bord X.
3.6.4. The visual metametric based at a point ξ ∈ ∂X. Our final metametric is supposed to generalize the Euclidean metric on the boundary of the halfspace model E. This metric should be thought of as “seen from the point ∞”.
Notation 3.6.17. If X is a hyperbolic metric space and ξ ∈ ∂X, then let
Eξ := bord X \ {ξ}.
Since we have not yet introduced a formula analogous to (3.5.2) for the Euclidean metric on ∂E \ {∞}, we will instead motivate the visual metametric based

at a point ξ ∈ ∂X by considering a sequence (wn )∞
1 in X converging to ξ, and
taking the limits of their visual metametrics.
In fact, Db,wn (y1 , y2 ) → 0 for every y1 , y2 ∈ Eξ . Some normalization is needed.

54

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES

Lemma 3.6.18. Fix o ∈ X, and suppose wn → ξ ∈ ∂X. Then for all y1 , y2 ∈ Eξ ,
bkwn k Db,wn (y1 , y2 ) −−→ b−[hy1 |y2 io −

P2

i=1 hyi |ξio ]

n,×

.

with −
→ if X is strongly hyperbolic.
n

Proof.
bkwn k Db,wn (y1 , y2 ) ≍× b−[hy1 |y2 iwn −kwn k]
≍× b−[hy1 |y2 io −

(by (3.6.4))

P2

−−→ b−[hy1 |y2 io −

i=1 hyi |wn io ]

P2

i=1 hyi |ξio ]

n,×

(by (k) of Proposition 3.3.3)

.

(by Lemma 3.4.22)

In each step, equality holds if X is strongly hyperbolic.



We can now construct the visual metametric based at a point ξ ∈ ∂X.
Proposition 3.6.19. For each o ∈ X and ξ ∈ ∂X, there exists a complete

metametric Db,ξ,o on Eξ satisfying

Db,ξ,o (y1 , y2 ) ≍× b−[hy1 |y2 io −

(3.6.5)

P2

i=1 hyi |ξio ]

,

with equality if X is strongly hyperbolic. The metametric Db,ξ,o is compatible with
the topology T ↿ Eξ ; moreover, a set S ⊆ Eξ is bounded in the metametric Db,ξ,o if
and only if ξ ∈
/ S.

Remark 3.6.20. The metric Db,ξ,o ↿ Eξ ∩ ∂X has been referred to in the

literature as the Hamenstädt metric.

Proof of Proposition 3.6.19. Let
Db,ξ,o (y1 , y2 ) = lim sup bkwk Dw (y1 , y2 )
w→ξ



kwn k
→ξ
:= sup lim sup b
Dwn (y1 , y2 ) : wn −
n→∞

n

Since the class of metametrics is closed under suprema and limits, it follows that
Db,ξ,o is a metametric. The asymptotic (3.6.5) follows from Lemma 3.6.18.
For the remainder of this proof, we write D = Db,o and Dξ = Db,ξ,o .
For all x ∈ Eξ ,
(3.6.6)

Dξ (o, x) ≍× b−[ho|xio−ho|ξio −hx|ξio ] = bhx|ξio ≍×

1
,
D(x, ξ)

with equality if X is strongly hyperbolic. It follows that for any set S ⊆ Eξ , the

function Dξ (o, ·) is bounded on S if and only if the function D(·, ξ) is bounded from
below on S. This demonstrates that S is bounded in the Dξ metametric if and only
if ξ ∈
/ S.

3.6. METRICS AND METAMETRICS ON bord X

55

Let (xn )∞
1 be a Cauchy sequence with respect to Dξ . Since D .× Dξ , it follows
that (xn )∞
1 is also Cauchy with respect to the metametric D, so it converges to a
point x ∈ bord X with respect to D. If x ∈ Eξ , then we have
Dξ (xn , x) ≍× bhxn |ξio +hx|ξio D(xn , x) −−→ b2hx|ξio 0 = 0.
n,×

On the other hand, if x = ξ, then the sequence (xn )∞
1 is unbounded in the Dξ
metametric, which contradicts the fact that it is Cauchy. Thus Dξ is complete.
Finally, given η ∈ (Eξ )refl = Eξ ∩ ∂X, consider the filters F1 and F2 generated
by the collections {BD (η, r) : r > 0} and {BDξ (η, r) : r > 0}, respectively. Since

D .× Dξ , we have F2 ⊆ F1 . Conversely, since BDξ (η, 1) is bounded in the Dξ
metametric, its closure does not contain ξ, and so the function h·|ξio is bounded

on this set. Thus D ≍×,η Dξ on BDξ (η, 1). Letting C be the implied constant of
the asymptotic, we have BDξ (η, min(r, 1)) ⊆ BD (η, Cr), which demonstrates that

F1 ⊆ F2 . Thus Dξ is compatible with the topology T ↿ Eξ .



From Lemma 3.6.18 and Proposition 3.6.19 it immediately follows that
bd(o,wn) Db,wn (y1 , y2 ) −−→ Db,ξ,o (y1 , y2 )

(3.6.7)

n,×

whenever (wn )∞
1 ∈ ξ.
Remark 3.6.21. It is not clear whether a result analogous to Proposition 3.6.13
holds for the metametric Db,ξ,o . A straightforward adaptation of the proof of Proposition 3.6.19 does not work, since
bkwn k Db,wn (x, y) = min(bkwn k d(x, y)), bkwn k Db,wn (x, y)
−−→ min(∞, Db,ξ,o (x, y))
n,×

= Db,ξ,o (x, y).

We finish this chapter by describing the relation between the visual metametric
based at ∞ and the Euclidean metric on the boundary of the half-space model E.
Proposition 3.6.22 (Cf. Figure 3.6.1). Let X = E = Eα , let o = (1, 0) ∈ X,

and fix x, y ∈ E∞ = E ∪ B. We have
(3.6.8)

De,∞,o (x, y) ≍× max(x1 , y1 , ky − xk),

with equality if x, y ∈ B = ∂E \ {∞}.

56

3. R-TREES, CAT(-1) SPACES, AND GROMOV HYPERBOLIC METRIC SPACES
∞

x
y

Figure 3.6.1. The Hamenstädt distance De,∞,o (x, y) between
two points x, y ∈ E is coarsely asymptotic to the maximum of
the following three quantities: x1 , y1 , and ky − xk. Equivalently,
De,∞,o (x, y) is coarsely asymptotic to the length of the shortest
path which both connects x and y and touches B.

Proof. First suppose that x, y ∈ E. By (h) of Proposition 3.3.3,



1
d(x, y) + B ∞ (o, x) + B ∞ (o, y)
De,∞,o (x, y) = exp
2
 


√
1
ky − xk2
x1 y1 exp
=
cosh−1 1 +
2
2x1 y1
(by (2.5.3) and Proposition 3.5.5)
s
ky − xk2
√
≍× x1 y1 1 +
2x1 y1
p
(since et/2 ≍× cosh(t))
p
x1 y1 + ky − xk2
=
√
≍× max( x1 y1 , ky − xk).

Since

√
x1 y1 ≤ max(x1 , y1 ), this demonstrates the . direction of (3.6.8). Since

y1 ≤ x1 + ky − xk and x1 ≤ y1 + ky − xk, we have

√
max(x1 , y1 ) .× max(min(x1 , y1 ), ky − xk) ≤ max( x1 y1 , ky − xk)

which demonstrates the reverse inequality. Thus (3.6.8) holds for x, y ∈ E; a

continuity argument demonstrates (3.6.8) for x, y ∈ E∞ .

3.6. METRICS AND METAMETRICS ON bord X

If x, y ∈ B, then

57

 


1
ky − xk2
cosh−1 1 +
a,b→0
2
2ab
s 

p
√
ky − xk2
(since lim et/2 / 2 cosh(t) = 1)
= lim ab 2 1 +
t→∞
a,b→0
2ab
s 

p
ky − xk2
2 ab +
= ky − xk2 = ky − xk.
= lim
a,b→0
2

De,∞,o (x, y) = lim

√
ab exp



Corollary 3.6.23 (Cf. [165, Fig. 5]). For x, y ∈ E, we have
ed(x,y) ≍×

max(x21 , y12 , ky − xk2 )
·
x1 y1

Proof. The result follows from

max(x21 , y12 , ky − xk2 ) ≍× De,∞,o (x, y)2 = exp d(x, y) + B ∞ (o, x) + B ∞ (o, y)
= x1 y1 ed(x,y)

which may be easily rearranged to complete the proof.



CHAPTER 4

More about the geometry of hyperbolic metric
spaces
In this chapter we discuss various topics regarding the geometry of hyperbolic
metric spaces, including metric derivatives, the Rips condition for hyperbolicity,
construction of geodesic rays and lines in CAT(-1) spaces, “shadows at infinity”,
and some functions which we call “generalized polar coordinates”. We start by
introducing some conventions to apply in the remainder of the paper.
4.1. Gromov triples
The following definition is made for convenience of notation:
Definition 4.1.1. A Gromov triple is a triple (X, o, b), where X is a hyperbolic
metric space, o ∈ X, and b > 1 is close enough to 1 to guarantee for every w ∈ X
the existence of a visual metametric Db,w via Proposition 3.6.8 above.
Notation 4.1.2. Let (X, o, b) be a Gromov triple. Given w ∈ X and ξ ∈ ∂X,

we will let Dw = Db,w be the metametric defined in Proposition 3.6.8, we will let
Dw = Db,w be the metric defined in Proposition 3.6.13, and we will let Dξ,w =
Db,ξ,w be the metametric defined in Proposition 3.6.19. If w = o, then we use the
further shorthand D = Do , D = Do , and Dξ = Dξ,o .

We will denote the diameter of a set S with respect to the metametric D by
Diam(S).
Convention 7. For the remainder of the paper, with the exception of Chapter
5, all statements should be assumed to be universally quantified over Gromov triples
(X, o, b) unless context indicates otherwise.
Convention 8. For the remainder of the paper, whenever we make statements
of the form “Let X = Y ”, where Y is a hyperbolic metric space, we implicitly want
to “beef up” X into a Gromov triple (X, o, b) whose underlying hyperbolic metric
space is Y . For general Y , this may be done arbitrarily, but if Y is strongly
hyperbolic, we want to set b = e, and if Y is an algebraic hyperbolic space, then
we want to set o = [(1, 0)], o = 0, or o = (1, 0) depending on whether Y is the
hyperboloid model H, the ball model B, or the half-space model E, respectively.
59

60

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

For example, when saying “Let X = H = H∞ ”, we really mean “Let X = H =
H∞ , let o = [(1, 0)], and let b = e.”
Convention 9. The term “Standard Case” will always refer to the finitedimensional situation where X = Hd for some 2 ≤ d < ∞.

4.2. Derivatives
4.2.1. Derivatives of metametrics. Let (Z, T ) be a perfect topological
space, and let D1 and D2 be two metametrics on Z. The metric derivative of
D1 with respect to D2 is the function D1 /D2 : Z → [0, ∞] defined by
D1 (z, w)
D1
(z) := lim
,
w→z
D2
D2 (z, w)
assuming the limit exists. If the limit does not exist, then we can speak of the
upper and lower derivatives; these will be denoted (D1 /D2 )∗ (z) and (D1 /D2 )∗ (z),
respectively. Note that the chain rule for metric derivatives takes the following
form:

D1 D2
D1
=
,
D3
D2 D3

assuming all limits exist.
We proceed to calculate the derivatives of the metametrics that were introduced
in Section 3.6:
Observation 4.2.1. Fix y1 , y2 ∈ bord X.
(i) For all w1 , w2 ∈ X, we have

(4.2.1)

Dw1 (y1 , y2 )
1
≍× b− 2 [By1 (w1 ,w2 )+By2 (w1 ,w2 )] .
Dw2 (y1 , y2 )

(ii) For all w ∈ X and ξ ∈ ∂X, we have
(4.2.2)

Dξ,w (y1 , y2 )
≍× b−[hy1 |ξiw +hy2 |ξiw ] .
Dw (y1 , y2 )

(iii) For all w1 , w2 ∈ X and ξ ∈ ∂X, we have
(4.2.3)

Dξ,w1 (y1 , y2 )
≍× bBξ (w1 ,w2 ) .
Dξ,w2 (y1 , y2 )

In each case, equality holds if X is strongly hyperbolic.
Proof. (i) follows from (g) of Proposition 3.3.3, while (ii) is immediate from
(3.6.4) and (3.6.5). (iii) follows from (3.6.7).
Combining with Lemma 3.4.22 yields the following:
Corollary 4.2.2. Suppose that bord X is perfect. Fix y ∈ bord X.



4.2. DERIVATIVES

61

(i) For all w1 , w2 ∈ X, we have

∗


Dw1
Dw1
(4.2.4)
(y) ≍× b− By (w1 ,w2 ) .
(y) ≍×
Dw2
Dw2 ∗
(ii) For all w ∈ X and ξ ∈ ∂X, we have

∗


Dξ,w
Dξ,w
(y) ≍×
(y) ≍× b−2hy|ξiw .
(4.2.5)
Dw
Dw ∗
(iii) For all w1 , w2 ∈ X and ξ ∈ ∂X, we have

∗


Dξ,w1
Dξ,w1
(y) ≍× bBξ (w1 ,w2 ) .
(y) ≍×
(4.2.6)
Dξ,w2
Dξ,w2 ∗
In each case, equality holds if X is strongly hyperbolic.
Remark 4.2.3. In case bord X is not perfect, (4.2.4) - (4.2.6) may be taken as
definitions. We will ignore the issue henceforth.
Combining Observation 4.2.1 with Corollary 4.2.2 yields the following:
Proposition 4.2.4 (Geometric mean value theorem). Fix y1 , y2 ∈ bord X.
(i) For all w1 , w2 ∈ X, we have

1/2
Dw1
Dw1
Dw1 (y1 , y2 )
≍×
(y1 )
(y2 )
.
Dw2 (y1 , y2 )
Dw2
Dw2

(ii) For all w ∈ X and ξ ∈ ∂X, we have

1/2
Dξ,w
Dξ,w (y1 , y2 )
Dξ,w
(y1 )
(y2 )
≍×
.
Dw (y1 , y2 )
Dw
Dw
(iii) For all w1 , w2 ∈ X and ξ ∈ ∂X, we have

1/2
Dξ,w1 (y1 , y2 )
Dξ,w1
Dξ,w1
≍×
(y1 )
(y2 )
.
Dξ,w2 (y1 , y2 )
Dξ,w2
Dξ,w2
In each case, equality holds if X is strongly hyperbolic.
4.2.2. Derivatives of maps. As before, let (Z, T ) be a perfect topological
space, and now fix just one metametric D on Z. For any map g : Z → Z, the
metric derivative of G is the function g ′ : Z → (0, ∞) defined by
g ′ (z) :=

D(g(z), g(w))
D◦g
(z) = lim
·
w→z
D
D(z, w)

If the limit does not exist, the upper and lower metric derivatives will be denoted
g′ and g′ , respectively.
Remark 4.2.5. To avoid confusion, in what follows g ′ will always denote the
derivative of an isometry g ∈ Isom(X) with respect to the metametric D = Db,o ,
rather than with respect to any other metametric.

62

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

Proposition 4.2.6. For all g ∈ Isom(X),
g′ (y) ≍× g′ (y) ≍× b− By (g

−1

(o),o)

∀y ∈ bord X

D(g(y1 ), g(y2 ))
1/2
∀y1 , y2 ∈ bord X,
≍× (g′ (y1 )g ′ (y2 ))
D(y1 , y2 )
with equality if X is strongly hyperbolic.
Proof. This follows from (i) of Corollary 4.2.2, (i) of Proposition 4.2.4, and
the fact that D ◦ g = Dg−1 (o) .

Corollary 4.2.7. For any distinct y1 , y2 ∈ Fix(g) ∩ ∂X we have
g ′ (y1 )g ′ (y2 ) ≍× 1,
with equality if X is strongly hyperbolic.
The next proposition shows the relation between the derivative of an isometry
g ∈ Isom(X) at a point ξ ∈ Fix(g) and the action on the metametric space (Eξ , Dξ ):
Proposition 4.2.8. Fix g ∈ Isom(X) and ξ ∈ Fix(g). Then for all y1 , y2 ∈ Eξ ,
Dξ (g(y1 ), g(y2 ))
1
≍× ′ ,
Dξ (y1 , y2 )
g (ξ)
with equality if X is strongly hyperbolic.
Proof.
Dξ (g(y1 ), g(y2 ))
=
Dξ (y1 , y2 )

Dξ,g−1 (o) (y1 , y2 )
Dξ,o (y1 , y2 )

≍× b− Bξ (o,g

≍× 1/g ′ (ξ).

−1

(o))

(by (4.2.3))
(by Proposition 4.2.6)


Remark 4.2.9. Proposition 4.2.8 can be interpreted as a geometric mean value
theorem for the action of g on the metametric space (Eξ , Dξ ). Specifically, it tells
us that the derivative of g on this metametric space is identically 1/g ′ (ξ).
Remark 4.2.10. If g ′ (ξ) = 1, then Proposition 4.2.8 tells us that the biLipschitz constant of g is independent of g, and that g is an isometry if X is
strongly hyperbolic. This special case will be important in Chapter 11.
Example 4.2.11. Suppose that X = E = Eα is the half-space model of a real
hyperbolic space, let B = ∂E \ {∞}, let g(x) = λT (x) + b be a similarity of B, and

consider the Poincaré extension gb ∈ Isom(E) defined in Observation 2.5.6. Clearly

4.2. DERIVATIVES

63

g+ = ∞

x

o
g−1 (x)
height(g−1 (x))

g−1 (o)
g−

Figure 4.2.1. The derivative of g at ∞ is equal to the reciprocal
of the dilatation ratio of g. In particular, ∞ is an attracting fixed
point if and only if g is expanding, and ∞ is a repelling fixed point
if and only if g is contracting.
g acts as a similarity on the metametric space (E∞ , D∞ ) in the following sense: For
b
all y1 , y2 ∈ E∞ ,

D∞ (g(y1 ), g(y2 )) = λD∞ (y1 , y2 ).

Comparing with Proposition 4.2.8 shows that g ′ (∞) = 1/λ.
4.2.3. The dynamical derivative. We can interpret Corollary 4.2.2 as saying that the metric derivative is well-defined only up to an asymptotic in a general
hyperbolic metric space (although it is perfectly well defined in a strongly hyperbolic metric space). Nevertheless, if ξ is a fixed point of the isometry g, then we
can iterate in order to get arbitrary accuracy.
Proposition 4.2.12. Fix g ∈ Isom(X) and ξ ∈ Fix(g). Then
1/n
1/n
g ′ (ξ) := lim (g n )′ (ξ)
= lim (g n )′ (ξ)
.
n→∞

n→∞

Furthermore

g′ (ξ) ≤ g ′ (ξ) ≤ g ′ (ξ).
The number g ′ (ξ) will be called the dynamical derivative of g at ξ.
Proof of Proposition 4.2.12. The limits converge due to the submultiplicativity and supermultiplicativity of the expressions inside the radicals, respectively. To see that they converge to the same number, note that by Corollary
4.2.2
lim

n→∞



(g n )′ (ξ)
(g n )′ (ξ)

1/n

≤ lim C 1/n = 1
n→∞

64

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

for some constant C independent of n.



Remark 4.2.13. Let βξ denote the Busemann quasicharacter of [48, p.14].
Then βξ is related to the dynamical derivative via the following formula: g ′ (ξ) =
b−βξ (g) .
Note that although the dynamical derivative is “well-defined”, it is not necessarily the case that the chain rule holds for any two g, h ∈ Stab(Isom(X); ξ) (although

it must hold up to a multiplicative coarse asymptotic). For a counterexample see
[48, Example 3.12]. Note that this counterexample includes the possibility of two
elementa g, h ∈ Stab(Isom(X); ξ) such that g ′ (ξ) = h′ (ξ) = 1 but (gh)′ (ξ) 6= 1. A

sufficient condition for the chain rule to hold exactly is given in [48, Corollary 3.9].
Despite the failure of the chain rule, the following “iteration” version of the
chain rule holds:
Proposition 4.2.14. Fix g ∈ Isom(X) and ξ ∈ Fix(g). Then
(g n )′ (ξ) = [g ′ (ξ)]n ∀n ∈ Z.
In particular
(4.2.7)

(g −1 )′ (ξ) =

1
g ′ (ξ)

·

Proof. The only difficulty lies in establishing (4.2.7):
1/n
1
(g −1 )′ (ξ) = lim (g −n )′ (ξ)
= exp1/b lim
B ξ (o, g n (o))
n→∞
n→∞ n
1
B ξ (g −n (o), o)
= exp1/b lim
n→∞ n


1
−n
= exp1/b − lim
B ξ (o, g (o))
n→∞ n
1
= ′ ·
g (ξ)

Combining with Corollary 4.2.7 yields the following:
Corollary 4.2.15. For any distinct y1 , y2 ∈ Fix(g) ∩ ∂X we have
g ′ (y1 )g ′ (y2 ) = 1.
We end this section with the following result relating the dynamical derivative
with the Busemann function:
Proposition 4.2.16. Fix g ∈ Isom(X) and ξ ∈ Fix(g). Then for all x ∈ X
and n ∈ Z,
B ξ (x, g −n (x)) ≍+ n logb g ′ (ξ).

4.3. THE RIPS CONDITION

65

with equality if X is strongly hyperbolic.
Proof. If x = o, then
b− Bξ (o,g

−n

(o))

≍× bBξ (g

−n

(o),o)

≍× (g n )′ (ξ)

(by Proposition 4.2.6)

≍× (g n )′ (ξ) = (g ′ (ξ))n .

For the general case, we note that
B ξ (x, g −n (x)) ≍+ B ξ (x, o) + B ξ (o, g −n (o)) + B ξ (g −n (o), g −n (x))
≍+ B ξ (x, o) + n logb g ′ (ξ) + B ξ (o, x)

≍+ n logb g ′ (ξ).



4.3. The Rips condition
In this section, in addition to assuming that X is a hyperbolic metric space (cf.
§4.1), we assume that X is geodesic. Recall (Section 3.2) that [x, y] denotes the
geodesic segment connecting two points x, y ∈ X.
Proposition 4.3.1.
(i) For all x, y, z ∈ X,
d(z, [x, y]) ≍+ hx|yiz .
(ii) (Rips’ thin triangles condition) For all x, y1 , y2 ∈ X and for any z ∈
[y1 , y2 ], we have

2

min d(z, [x, yi ]) ≍+ 0.
i=1

In fact, the thin triangles condition is equivalent to hyperbolicity; see e.g. [39,
Proposition III.H.1.22].
Proof.
(i) By the intermediate value theorem, there exists w ∈ [x, y] such that
hx|ziw = hy|ziw . Applying Gromov’s inequality gives hx|ziw = hy|ziw .+
hx|yiw = 0. Now (k) of Proposition 3.3.3 shows that
d(z, [x, y]) ≤ d(z, w) .+ hx|yiz .
The other direction is immediate, since for each w ∈ [x, y], we have

hx|yiw = 0, and so (d) of Proposition 3.3.3 gives hx|yiw ≤ d(z, w).

66

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

x

y
տ
z

≍+ hx, yiz

Figure 4.3.1. An illustration of Proposition 4.3.1(i).

(ii) This is immediate from (i), Gromov’s inequality, and the equation
hy1 |y2 iz = 0.

The next lemma demonstrates the correctness of the intuitive notion that if
two points are close to each other, then the geodesic connecting them should not
be very large.
Lemma 4.3.2. Fix x1 , x2 ∈ bord X. We have
Diam([x1 , x2 ]) ≍× D(x1 , x2 ).
Proof. It suffices to show that if y ∈ [x1 , x2 ], then
D(y, {x1 , x2 }) .× D(x1 , x2 ).
Indeed, by the thin triangles condition, we may without loss of generality suppose
that d(y, [o, x1 ]) ≍+ 0. Write d(y, z) ≍+ 0 for some z ∈ [o, x1 ]. Then
D(x1 , y) ≍× D(x1 , z) = e−kzk ≍× e−kyk ≤ e−d(o,[x1,x2 ])
≍× e−hx1 |x2 io
≍× D(x1 , x2 ).


4.4. GEODESICS IN CAT(-1) SPACES

67

4.4. Geodesics in CAT(-1) spaces
Observation 4.4.1. Any isometric embedding π : [t, ∞) → X extends uniquely

to a continuous map π : [t, ∞] → bord X. Similarly, any isometric embedding π :

(−∞, +∞) → X extends uniquely to a continuous map π : [−∞, +∞] → bord X.
Abusing terminology, we will also call the extended maps “isometric embeddings”.
Definition 4.4.2. Fix x ∈ X and ξ, η ∈ ∂X.
• A geodesic ray connecting x and ξ is the image of an isometric embedding
π : [0, ∞] → X satisfying

π(0) = x,

π(∞) = ξ.

• A geodesic line or bi-infinite geodesic connecting ξ and η is the image of
an isometric embedding π : [−∞, +∞] → X satisfying
π(−∞) = ξ,

π(+∞) = η.

When we do not wish to distinguish between geodesic segments (cf. Section 3.2),
geodesic rays, and geodesic lines, we shall simply call them geodesics. For x, y ∈
bord X, any geodesic connecting x and y will be denoted [x, y].
Notation 4.4.3. Extending Notation 3.1.6, if [x, ξ] is the image of the isometric
embedding π : [0, ∞] → X, then for t ∈ [0, ∞] we let [x, ξ]t = π(t), i.e. [x, ξ]t is the
unique point on the geodesic ray [x, ξ] such that d(x, [x, ξ]t ) = t.
The main goal of this section is to prove the following:
Proposition 4.4.4. Suppose that X is a complete CAT(-1) space. Then:
(i) For any two distinct points x, y ∈ bord X, there is a unique geodesic [x, y]
connecting them.
∞
(ii) Suppose that (xn )∞
1 and (yn )1 are sequences in bord X which converge
to points xn → x ∈ bord X and yn → y ∈ bord X, with x 6= y. Then

[xn , yn ] → [x, y] in the Hausdorff metric on (bord X, D). If x = y, then
[xn , yn ] → {x} in the Hausdorff metric.

Definition 4.4.5. A hyperbolic metric space X satisfying the conclusion of
Proposition 4.4.4 will be called regularly geodesic.
Remark 4.4.6. The existence of a geodesic connecting any two points in bord X
was proven in [42, Proposition 0.2] under the weaker hypothesis that X is a Gromov
hyperbolic complete CAT(0) space. However, this weaker hypothesis does not imply
the uniqueness of such a geodesic, nor does it imply (ii) of Proposition 4.4.4, as
shown by the following example:

68

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

Example 4.4.7 (A proper and uniquely geodesic hyperbolic CAT(0) space
which is not regularly geodesic). Let
X = {x ∈ R2 : x2 ∈ [0, 1]}
be interpreted as a subspace of R2 with the usual metric. Then X is hyperbolic,
proper, and uniquely geodesic, but is not regularly geodesic.
Proof. It is hyperbolic since it is roughly isometric to R. It is uniquely geodesic since it is a convex subset of R2 . It is proper because it is a closed subset of
R2 . It is not regularly geodesic because if we write ∂X = {ξ+ , ξ− }, then the two

points ξ+ and ξ− have infinitely many distinct geodesics connecting them: for each
t ∈ [0, 1], R × {t} is a geodesic connecting ξ+ and ξ− .

The proof of Proposition 4.4.4 will proceed through several lemmas, the first
of which is as follows:
Lemma 4.4.8. Fix ε > 0. There exists δ = δX (ε) > 0 such that if ∆ =
∆(x, y1 , y2 ) is a geodesic triangle in X satisfying
D(y1 , y2 ) ≤ δ,

(4.4.1)

then for all t ∈ [0, min2i=1 d(x, yi )], if zi = [x, yi ]t , then
D(z1 , z2 ) ≤ ε.

(4.4.2)

Proof. We prove the assertion first for X = H2 and then in general:
1
n

If X = H2 : Let ε > 0, and by contradiction, suppose that for each δ =
(n) (n) (n) (n)
(x(n) , y1 , y2 , z1 , z2 )

> 0 there

exists a 5-tuple
satisfying the hypotheses but
not the conclusion of the theorem. Since bord H2 is compact, there exists
a convergent subsequence
(nk )

(x(nk ) , y1

(nk )

, y2

(nk )

, z1

(nk )

, z2

) → (x, y1 , y2 , z1 , z2 ) ∈ bord H2

5

.

Taking the limit of (4.4.1) as k → ∞ shows that D(y1 , y2 ) = 0, so y1 = y2 .
Conversely, taking the limit of (4.4.2) shows that D(z1 , z2 ) ≥ ε > 0, so
z1 6= z2 . Write y = y1 = y2 .
We will take for granted that Proposition 4.4.4 holds when X = H2 .

(This can be proven using the explicit form of geodesics in this space.) It
follows that zi ∈ [x, y] if x 6= y, and zi = x if x = y. The second case is
clearly a contradiction, so we assume that x 6= y.
(n )
(n )
Writing zi k = [x(nk ) , yi k ]tk , we observe that
(nk )

tk − kx(nk ) k = hzi

(nk )

|yi

(nk )

io − hzi

−
→ hzi |yio − hzi |xio − hx|yio .
k

(nk )

|x(nk ) io − hx(nk ) |yi

io

4.4. GEODESICS IN CAT(-1) SPACES

69

y1
z1
w1
o

y2
w2

z2

x

Figure 4.4.1. The triangle ∆(x, y1 , y2 ).

Since the left hand side is independent of i, so is the right hand side. But
the function
z 7→ hz|yio − hz|xio − hx|yio
is an isometric embedding from [x, y] to [−∞, +∞]; it is therefore injective.
Thus z1 = z2 , a contradiction.
In general: Let ε > 0, and fix εe > 0 to be determined, depending on ε. Let δe = δH2 (e
ε),
e Now suppose that
and fix δ > 0 to be determined, depending on δ.

∆ = ∆(x, y1 , y2 ) is a geodesic triangle in X satisfying (4.4.1), fix t ≥ 0, and
let zi = [x, yi ]t . To complete the proof, we must show that D(z1 , z2 ) ≤ ε.
By contradiction suppose not, i.e. suppose that D(z1 , z2 ) > ε. Then
D(x, zi ) > ε/2 for some i = 1, 2; without loss of generality suppose

D(x, z1 ) > ε/2. By Proposition 4.3.1 this implies d(o, [x, z1 ]) ≍+,ε 0; fix
w1 ∈ [x, z1 ] with kw1 k ≍+,ε 0. Let s = d(x, w1 ) ≤ t, and let w2 = [x, z2 ]s .

(See Figure 4.4.1.)

Now let ∆ = ∆(x, y1 , y2 ) be a comparison triangle for ∆(x, y1 , y2 ), and
let z1 , z2 , w1 , w2 be the corresponding comparison points. Note that zi =
[x, yi ]t and wi = [x, yi ]s . Without loss of generality, suppose that w1 =
oH . Then ky2 k ≤ kw1 k + d(w1 , y2 ) ≍+,ε d(oH , y2 ), and so hy1 |y2 io .+,ε
hy1 |y2 ioH , and thus

D(y1 , y2 ) .×,ε D(y1 , y2 ) ≤ δ.
Setting δ equal to δe divided by the implied constant, we have
D(y1 , y2 ) ≤ δe = δH (e
ε).

(4.4.3)

Thus D(z1 , z2 ) ≤ εe and D(w1 , w2 ) ≤ εe.
– If d(z1 , z2 ) ≤ εe, then the CAT(-1) inequality finishes the proof (as
long as εe ≤ ε). Thus, suppose that
D(z1 , z2 ) ≤ εe.

70

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

(4.4.4)

ε), a contradiction for
– If D(w1 , w2 ) ≤ εe, then 0 = hw1 |w2 ioH ≥ − log(e
εe sufficiently small. Thus, suppose that
d(w1 , w2 ) ≤ εe.

ε). Applying (4.4.4) gives
By (4.4.3), we have d(oH , zi ) ≥ − log(e
hzi |yi iwi = d(wi , zi ) = d(wi , zi ) &+ − log(e
ε).
Applying (4.4.4), the coarse asymptotic kw1 k ≍+,ε 0, and the CAT(-1)
inequality, we have
hzi |yi io &+,ε − log(e
ε),
and thus D(zi , yi ) .×,ε εe. Using the triangle inequality together with the

assumption D(y1 , y2 ) ≤ δ, we have

D(z1 , z2 ) .×,ε max(δ, εe).

Setting εe equal to ε divided by the implied constant, and decreasing δ if
necessary, completes the proof.



Notation 4.4.9. If the map π : [t, s] → X is an isometric embedding, then the

map π
e : [−∞, +∞] → X is defined by the equation
π
e(r) = π(t ∨ r ∧ s).

Corollary 4.4.10. If ε, δ, and ∆(x, y1 , y2 ) are as in Lemma 4.4.8, and if
π1 : [t, s1 ] → [x, y1 ] and π2 : [t, s2 ] → [x, y2 ] are isometric embeddings, then
D(e
π1 (r), π
e2 (r)) .× ε ∀r ∈ [−∞, +∞].

Proof. If r ≤ t, then π
e1 (r) = x = π
e2 (r). If t ≤ r ≤ min2i=1 si , then π
ei (r) =

[x, yi ]r−t , allowing us to apply Lemma 4.4.8 directly. Finally, suppose r ≥ r0 :=

min2i=1 si . Without loss of generality suppose that s1 ≤ s2 , so that r0 = s1 .
Applying the previous case to r0 , we have
D(y1 , w2 ) ≤ ε,

where w2 = π2 (s1 ). Now π
e1 (r) = y1 , and π
e2 (r) ∈ [w2 , y2 ], so Lemma 4.3.2 completes the proof.



∞
Lemma 4.4.11. Suppose that (xn )∞
1 and (yn )1 are sequences in X which con-

verge to points xn → x ∈ bord X and yn → y ∈ bord X, with x 6= y. Then
there exists a geodesic [x, y] connecting x and y such that [xn , yn ] → [x, y] in the

Hausdorff metric. If x = y, then [xn , yn ] → {x} in the Hausdorff metric.

4.4. GEODESICS IN CAT(-1) SPACES

71

Proof. We observe first that if x = y, then the conclusion follows immediately
from Lemma 4.3.2. Thus we assume in what follows that x 6= y.
For any pair p, q ∈ X, we define the standard parameterization of the geodesic

[p, q] to be the unique isometry π : [−ho|qip , ho|piq ] → [p, q] sending −ho|qip to p
and ho|piq to q. For each n let πn : [tn , sn ] → [xn , yn ] be the standard parameterization, and for each m, n ∈ N let πm,n : [tm,n , sm,n ] → [xn , ym ] be the standard
parameterization. Let π
en : [−∞, +∞] → [xn , yn ] and π
em,n : [−∞, +∞] → [xn , ym ]
be as in Notation 4.4.9. Note that

tn − tm,n = ho|ym ixn − ho|yn ixn = hxn |yn io − hxn |ym io −−→ hx|yio − hx|yio = 0.
m,n

(We have hx|yio < ∞ since x 6= y.) Thus
D(e
πn (r), π
en (r − tn + tm,n )) ≤ d(e
πn (r), π
en (r − tn + tm,n )) ≤ |tn − tm,n | −
→ 0.
n

Here and below, the limit converges uniformly for r ∈ [−∞, +∞]. On the other
hand, Corollary 4.4.10 implies that
D(e
πn (r − tn + tm,n ), π
em,n (r)) −−→ 0,
m,n

so the triangle inequality gives

D(e
πn (r), π
em,n (r)) −−→ 0.
m,n

A similar argument shows that

D(e
πm,n (r), π
em (r)) −−→ 0,
m,n

so the triangle inequality gives

D(e
πn (r), π
em (r)) −−→ 0,
m,n

i.e. the sequence of functions (e
πn )∞
1 is uniformly Cauchy. Since (bord X, D) is
complete, they converge uniformly to a function π
e : [−∞, +∞] → X.
Clearly, [xn , yn ] = π
en ([−∞, +∞]) → π
e([−∞, +∞]) in the Hausdorff metric.

We claim that π
e([−∞, +∞]) is a geodesic connecting x and y. Indeed,
tn −
→ t := hx|yio − kxk and sn −
→ s := kyk − hx|yio .
n

n

For all t < r1 < r2 < s, we have tn < r1 < r2 < sn for all sufficiently large n, which
implies that
d(e
π (r1 ), π
e(r2 )) = lim d(e
πn (r1 ), π
en (r2 )) = lim (r2 − r1 ) = r2 − r1 ,
n→∞

n→∞

i.e. π
e ↿ (t, s) is an isometric embedding. Since π
e is continuous (being the uniform
limit of continuous functions), π := π
e ↿ [t, s] is also an isometric embedding. A
similar argument shows that π
e(r) = π(t) for all r ≤ t, and π
e(r) = π(s) for all r ≥ s;

72

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

thus π
e ([−∞, +∞]) = π([t, s]) is a geodesic. To complete the proof, we must show

that π(t) = x and π(s) = y. Indeed,

π(t) = π
e (−∞) = lim π
en (−∞) = lim xn = x,
n→∞

n→∞

and a similar argument shows that π(s) = y. Thus the geodesic π([t, s]) connects
x and y.

Using Lemma 4.4.11, we prove Proposition 4.4.4.
Proof of Proposition 4.4.4.
(i) Given distinct points x, y ∈ bord X, we may find sequences X ∋ xn → x
and X ∋ yn → y. Applying Lemma 4.4.11 proves the existence of a
geodesic connecting x and y. To show uniqueness, suppose that [x, y]1
and [x, y]2 are two geodesics connecting x and y. Fix sequences [x, y]1 ∋
(1)

(2)

(1)

(2)

xn → x, [x, y]2 ∋ xn → x, [x, y]1 ∋ yn → y,and [x, y]2 ∋ yn → y. By
considering the intertwined sequences
(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

x1 , x1 , x2 , x2 , . . .
and
y1 , y1 , y2 , y2 , . . .
(1)

(1)

(2)

(2)

∞
Lemma 4.4.11 shows that both sequences ([xn , yn ])∞
1 and ([xn , yn ])1
converge in the Hausdorff metric to a common geodesic [x, y]. But clearly

the former tend to [x, y]1 , and the latter tend to [x, y]2 ; we must have
[x, y]1 = [x, y]2 .
(ii) Suppose that bord X ∋ xn → x and bord X ∋ yn → y. For each n, choose
xn , xn ), D(b
yn , yn ) ≤ 1/n. Then x
bn → x
x
bn , ybn ∈ [xn , yn ] ∩ X such that D(b
and ybn → y, so by Lemma 4.4.11 we have [b
xn , ybn ] → [x, y] in the Hausdorff
metric, or [b
xn , ybn ] → {x} if x = y. To complete the proof it suffices to show
that the Hausdorff distance between [xn , yn ] and [b
xn , ybn ] tends to zero as

n tends to infinity. Indeed, [b
xn , ybn ] ⊆ [xn , yn ], and for each z ∈ [xn , yn ],
either z ∈ [xn , x
bn ], z ∈ [b
xn , ybn ], or z ∈ [b
yn , yn ]. In the first case, Lemma

4.3.2 shows that D(z, [b
xn , ybn ]) ≤ D(z, x
bn ) .× D(xn , x
bn ) ≤ 1/n → 0; the
third case is treated similarly.



Having completed the proof of Proposition 4.4.4, in the remainder of this section
we prove that a version of the CAT(-1) equality holds for ideal triangles.
Definition 4.4.12. A geodesic triangle ∆ = ∆(x, y, z) consists of three distinct
points x, y, z ∈ bord X together with the geodesics [x, y], [y, z], and [z, x].

4.5. THE GEOMETRY OF SHADOWS

73

A geodesic triangle ∆ = ∆(x, y, z) is called a comparison triangle for ∆ if
hx|yiz = hx|yiz , etc.
For any point p ∈ [x, y], its comparison point is defined to be the unique point
p ∈ [x, y] such that

hx|zip − hy|zip = hx|zip − hy|zip .

We say that the geodesic triangle ∆ satisfies the CAT(-1) inequality if for all points
p, q ∈ ∆ and for any comparison points p, q ∈ ∆, we have d(p, q) ≤ d(p, q).
It should be checked that these definitions are consistent with those given in
Section 3.2.
Proposition 4.4.13. Any geodesic triangle (including ideal triangles) satisfies
the CAT(-1) inequality.
Proof. Let ∆ = ∆(x, y, z) be a geodesic triangle, and fix p, q ∈ ∆. Choose

sequences xn → x, yn → y, and zn → z. By Proposition 4.4.4, we have ∆n =
∆(xn , yn , zn ) → ∆ in the Hausdorff metric, so we may choose pn , qn ∈ ∆n so that
pn → p, qn → q. For each n, let ∆n = ∆(xn , yn , z n ) be a comparison triangle for
∆n . Without loss of generality, we may assume that

(4.4.5)

o ∈ [xn , y n ] and hxn |z n io = hy n |z n io ≍+ 0.

By extracting a convergent subsequence, we may without loss of generality assume
that xn → x, y n → y, and z n → z for some points x, y, z ∈ bord H2 . By (4.4.5),
the points x, y, z are distinct. Thus ∆ = ∆(x, y, z) is a geodesic triangle, and is in
fact a comparison triangle for ∆. If p, q are comparison points for p, q, then pn → p
and q n → q. It follows that
d(p, q) = lim d(pn , qn ) ≤ lim d(pn , q n ) = d(p, q).
n→∞

n→∞



4.5. The geometry of shadows
4.5.1. Shadows in regularly geodesic hyperbolic metric spaces. Suppose that X is regularly geodesic. For each z ∈ X we consider the relation

πz ⊆ X × ∂X defined by

(x, ξ) ∈ πz ⇔ x ∈ [z, ξ]

(see Definition 4.4.2 for the definition of [z, ξ]). Note that if X is an algebraic
hyperbolic space, then the relation πz is a function when restricted to X \ {z}; in

74

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

πz B(x, σ)
B(x, σ)

σ



x

z

Figure 4.5.1. The set πz (B(x, σ)). Although this set is not equal
to Shadz (x, σ), they are approximately the same in regularly geodesic spaces by Corollary 4.5.5. In our drawings, we will draw the
set πz (B(x, σ)) to indicate the set Shadz (x, σ) (since the latter is
hard to draw).

particular, for x ∈ B = Bα
F with x 6= 0 we have
x
·
π0 (x) =
kxk

However, in general the relation πz is not necessarily a function; R-trees provide a
good counterexample. The reason is that in an R-tree, there may be multiple ways
to extend a geodesic segment to a geodesic ray.
For any set S, we define its shadow with respect to the light source z to be the
set
πz (S) := {ξ ∈ ∂X : ∃x ∈ S (x, ξ) ∈ πz }.
4.5.2. Shadows in hyperbolic metric spaces. In regularly geodesic hyper-

bolic metric spaces, it is particularly useful to consider πz (B(x, σ)) where x ∈ X

and σ > 0. We would like to have an analogue for this set in the Gromov hyperbolic
setting.
Definition 4.5.1. For each σ > 0 and x, z ∈ X, let
Shadz (x, σ) = {η ∈ ∂X : hz|ηix ≤ σ}.
We say that Shadz (x, σ) is the shadow cast by x from the light source z, with
parameter σ. For shorthand we will write Shad(x, σ) = Shado (x, σ).

4.5. THE GEOMETRY OF SHADOWS

75

The relation between πz (B(x, σ)) and Shadz (x, σ) in the case where X is a
regularly geodesic hyperbolic metric space will be made explicit in Corollary 4.5.5
below.
Let us establish up front some geometric properties of shadows.
Observation 4.5.2. For each x, z ∈ X and σ > 0 the set Shadz (x, σ) is closed.
Proof. This follows directly from Lemma 3.4.23.



Observation 4.5.3. If η ∈ Shadz (x, σ), then
hx|ηiz ≍+,σ d(z, x).
Proof. Follows directly from (b) of Proposition 3.3.3 together with the definition of Shadz (x, σ).



Lemma 4.5.4 (Intersecting Shadows Lemma). For each σ > 0, there exists
τ = τσ > 0 such that for all x, y, z ∈ X satisfying d(z, y) ≥ d(z, x) and Shadz (x, σ)∩

Shadz (y, σ) 6= , we have
(4.5.1)

Shadz (y, σ) ⊆ Shadz (x, τ )

and
(4.5.2)

d(x, y) ≍+,σ d(z, y) − d(z, x).

Proof. Fix η ∈ Shadz (x, σ) ∩ Shadz (y, σ), so that by Observation 4.5.3
hx|ηiz ≍+,σ d(z, x) and hy|ηiz ≍+,σ d(z, y) ≥ d(z, x).
Gromov’s inequality along with (c) of Proposition 3.3.3 then gives
(4.5.3)

hx|yiz ≍+,σ d(z, x).

Rearranging yields (4.5.2). In order to show (4.5.1), fix ξ ∈ Shadz (y, σ), so that

hy|ξiz ≍+,σ d(z, y) ≥ d(z, x). Gromov’s inequality and (4.5.3) then give
hx|ξiz ≍+,σ d(z, x),
i.e. ξ ∈ Shadz (x, τ ) for some τ > 0 sufficiently large (depending on σ).



Corollary 4.5.5. Suppose that X is regularly geodesic. For every σ > 0, there
exists τ = τσ > 0 such that for any x, z ∈ X we have
(4.5.4)

πz (B(x, σ)) ⊆ Shadz (x, σ) ⊆ πz (B(x, τ )).

Proof. Suppose ξ ∈ πz (B(x, σ)). Then there exists a point y ∈ B(x, σ)∩[z, ξ].
By (d) of Proposition 3.3.3
hz|ξix ≤ hz|ξiy + d(x, y) ≤ hz|ξiy + σ = σ,

76

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

Shadz (x, σ)

B(x, σ)

Shadz (y, σ)
B(y, σ)

z

Figure 4.5.2. In this figure, d(z, y) ≥ d(z, x) and Shadz (x, σ) ∩
Shadz (y, σ) 6= . The Intersecting Shadows Lemma (Lemma 4.5.4)
provides a τσ > 0 such that the shadow cast from z about B(x, τσ )
will capture Shadz (y, σ).
i.e. ξ ∈ Shadz (x, σ). This demonstrates the first inclusion of (4.5.4). On the other

hand, suppose that ξ ∈ Shadz (x, σ). Let y ∈ [z, ξ] be the unique point so that
d(z, y) = d(z, x). Clearly ξ ∈ Shadz (y, σ), so Shadz (x, σ) ∩ Shadz (y, σ) 6= ; by the

Intersecting Shadows Lemma 4.5.4 we have

d(x, y) ≍+,σ B ξ (y, x) = 0
i.e. d(x, y) ≤ τ for some τ = τσ > 0 depending only on σ. Then y ∈ B(x, τ ) ∩ [z, ξ],
which implies that ξ = πz (y) ∈ πz (B(x, τ )). This finishes the proof.

Lemma 4.5.6 (Bounded Distortion Lemma). Fix σ > 0. Then for every g ∈
Isom(X) and for every y ∈ Shadg−1 (o) (o, σ) we have
(4.5.5)

g′ (y) ≍×,σ b−kgk .

Moreover, for every y1 , y2 ∈ Shadg−1 (o) (o, σ), we have
(4.5.6)

D(g(y1 ), g(y2 ))
≍×,σ b−kgk .
D(y1 , y2 )
−1

−1

Proof. We have g ′ (y) ≍× bBy (o,g (o)) ≍× b2hg (o)|yio −kgk ≍×,σ b−kgk , giving
(4.5.5). Now (4.5.6) follows from (4.5.5) and the geometric mean value theorem
(Proposition 4.2.4).



4.5. THE GEOMETRY OF SHADOWS

77

σ
o

z
∂X \ Shadz (o, σ)

Figure 4.5.3. The Big Shadows Lemma 4.5.7 tells us that for any
ε > 0, we may choose σ > 0 sufficiently large so that Diam(∂X \
Shadz (o, σ)) ≤ ε for every z ∈ X.
Lemma 4.5.7 (Big Shadows Lemma). For every ε > 0, for every σ > 0 sufficiently large (depending on ε), and for every z ∈ X, we have
(4.5.7)

Diam(∂X \ Shadz (o, σ)) ≤ ε.

Proof. If ξ, η ∈ ∂X \ Shadz (o, σ), then hz|ξio > σ and hz|ηio > σ. Thus by

Gromov’s inequality we have

hξ|ηio &+ σ.

Exponentiating gives D(ξ, η) .× b

−σ

. Thus

Diam(∂X \ Shadz (o, σ)) .× b−σ −
→ 0,
σ

and the convergence is uniform in z.



Lemma 4.5.8 (Diameter of Shadows Lemma). For all σ > 0 sufficiently large,
we have for all g ∈ Isom(X) and for all z ∈ X
(4.5.8)

Diamz (Shadz (g(o), σ)) .×,σ b−d(z,g(o)) ,

with ≍ if #(∂X) ≥ 3. Moreover, for every C > 0 there exists σ > 0 such that
(4.5.9)

Bz (x, Ce−d(z,x) ) ⊆ Shadz (x, σ) ∀x, z ∈ X.

78

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

Shadz (g(o), σ)
σ
g(o)
x



B(g(o), σ)
z

Figure 4.5.4. The Diameter of Shadows Lemma 4.5.8 says that
the diameter of Shad(g(o), σ) is coarsely asymptotic to b−kgk .
Proof. Let x = g(o). For any ξ, η ∈ Shadz (x, σ), we have
Dz (ξ, η) ≍× b−hξ|ηiz .×

b− min(hx|ξiz ,hx|ηiz ) .×,σ b−d(z,x)

which demonstrates (4.5.8).
Now let us prove the converse of (4.5.8), assuming #(∂X) ≥ 3. Fix ξ1 , ξ2 , ξ3 ∈

∂X, let ε = mini6=j D(ξi , ξj )/2, and fix σ > 0 large enough so that (4.5.7) holds for
every z ∈ X. By (4.5.7) we have
Diam(∂X \ Shadg−1 (z) (o, σ)) ≤ ε,
and thus


# i = 1, 2, 3 : ξi ∈ Shadg−1 (z) (o, σ) ≥ 2.

Without loss of generality suppose that ξ1 , ξ2 ∈ Shadg−1 (z) (o, σ). By applying g,
we have g(ξ1 ), g(ξ2 ) ∈ Shadz (x, σ). Then
Diamz (Shadz (x, σ)) ≥ Dz (g(ξ1 ), g(ξ2 ))
≍× b−hg(ξ1 )|g(ξ2 )iz = b−hξ1 |ξ2 ig−1 (z)
&× b−hξ1 |ξ2 io b−kg

−1

(z)k

≍×,ξ1 ,ξ2 b−d(z,x) .

Finally, given y ∈ Bz (x, Cb−d(z,x) ), we have
hx|yiz &+ − logb (Cb−kxk ) ≍+ d(z, x)
and thus hz|yix ≍+ 0, demonstrating (4.5.9).



4.6. GENERALIZED POLAR COORDINATES

79

x
kxk

∡(x)

x1

0

Figure 4.6.1. The quantities kxk and ∡(x) can be interpreted as
“polar coordinates” of x.
4.6. Generalized polar coordinates
Suppose that X = E = Eα is the half-space model of a real hyperbolic space.
Fix a point x ∈ E, and consider the numbers kxk and ∡(x) := cos−1 (x1 /kxk),
i.e. the radial and unsigned angular coordinates of x. (The angular coordinate is
computed with respect to the ray {(t, 0) : t ∈ [0, ∞)}; cf. Figure 4.6.1.) These
“polar coordinates” of x do not completely determine x, but they are enough to
compute certain important quantities depending on x, e.g. dE (o, x), B ∞ (o, x), and
B 0 (o, x). (We omit the details.) In this section we consider a generalization, in a

loose sense, of these coordinates to an arbitrary hyperbolic metric space.
Let us note that the isometries of E which preserve the polar coordinate functions defined above are exactly those of the form Tb where T ∈ O(E). Equivalently,
these are the members of Isom(E) which preserve 0, o = (1, 0), and ∞. This suggests that our “coordinate system” is fixed by choosing a point in E and two distinct
points in ∂E.

We now return to the general case of §4.1. Fix two distinct points ξ1 , ξ2 ∈ ∂X.
Definition 4.6.1. The generalized polar coordinate functions are the functions
r = rξ1 ,ξ2 ,o and θ = θξ1 ,ξ2 ,o : X → R defined by

1
[B ξ1 (x, o) − B ξ2 (x, o)]
2
1
θ(x) = [B ξ1 (x, o) + B ξ2 (x, o)] ≍+ hξ1 |ξ2 ix − hξ1 |ξ2 io .
2
r(x) =

The connection between generalized polar coordinates and classical polar coordinates is given in Proposition 4.6.4 below. For now, we list some geometrical
facts about generalized polar coordinates. Our first lemma says that the hyperbolic

80

4. MORE ABOUT THE GEOMETRY OF HYPERBOLIC METRIC SPACES

distance from a point to the origin is essentially the sum of the “radial” distance
and the “angular” distance.
Lemma 4.6.2. For all x ∈ X we have
2

kxk ≍+,o,ξ1 ,ξ2 max B ξi (x, o) = |r(x)| + θ(x).
i=1

Proof. The equality is trivial, so we concentrate on the asymptotic. The &
direction follows directly from (f) of Proposition 3.3.3. On the other hand, by
Gromov’s inequality
2

2

i=1

i=1

kxk − max B ξi (x, o) ≍+ 2 minhx|ξi io .+ 2hξ1 |ξ2 io ≍+,o,ξ1 ,ξ2 0.

Our next lemma describes the effect of isometries on generalized polar coordinates.
Lemma 4.6.3. Fix g ∈ Isom(X) such that ξ1 , ξ2 ∈ Fix(g). For all x ∈ X we

have

(4.6.1)

r(g(x)) ≍+ r(x) + logb g ′ (ξ1 ) = r(x) − logb g ′ (ξ2 )

(4.6.2)

θ(g(x)) ≍+ θ(x),

with equality if X is strongly hyperbolic. The implied constants are independent of
g, ξ1 , and ξ2 .
Proof.

 
B ξ1 (g(x), o) − B ξ2 (g(x), o) − B ξ1 (x, o) + B ξ2 (x, o)

 

=
B ξ1 (x, g −1 (o)) − B ξ2 (x, g −1 (o)) − B ξ1 (x, o) − B ξ2 (x, o)

2[r(g(x)) − r(x)] =



≍+ B ξ1 (o, g −1 (o)) − B ξ2 (o, g −1 (o))
≍+ logb g ′ (ξ1 ) − logb g ′ (ξ2 ).

(by Proposition 4.2.16)

Now (4.6.1) follows from Corollary 4.2.15.
On the other hand, by (g) of Proposition 3.3.3,

 

θ(g(x)) − θ(x) ≍+ hξ1 |ξ2 ig(x) − hξ1 |ξ2 io − hξ1 |ξ2 ix − hξ1 |ξ2 io
= hg −1 (ξ1 )|g −1 (ξ2 )ix − hξ1 |ξ2 ix = 0,

proving (4.6.2).



We end this chapter by describing the relation between generalized polar coordinates and classical polar coordinates.

4.6. GENERALIZED POLAR COORDINATES

81

Proposition 4.6.4. If X = E, o = (1, 0), ξ1 = 0, and ξ2 = ∞, then
r(x) = log kxk
θ(x) = − log(x1 /kxk) = − log cos(∡(x)).
Thus the notations r and θ are slightly inaccurate as they really represent
the logarithm of the radius and the negative logarithm of the cosine of the angle,
respectively.
Proof of Proposition 4.6.4. We consider first the case kxk = 1. Let us set

g(y) = y/kyk2 , and note that g ∈ Isom(E), g(o) = o, and g(ξi ) = ξ3−i . On the
other hand, since kxk = 1 we have g(x) = x, and so
B ξ1 (x, o) = B g(ξ1 ) (g(x), g(o)) = B ξ2 (x, o).

It follows that r(x) = 0 and θ(x) = B ξ2 (x, o) = B ∞ (x, o). By Proposition 3.5.5, we

have B ∞ (x, o) = − log(x1 /o1 ) = − log(x1 /kxk) = − log cos(∡(x)).
The general case follows upon applying Lemma 4.6.3 to maps of the form
gλ (x) = λx, λ > 0.



CHAPTER 5

Discreteness
Let X be a metric space. In this chapter we discuss several different notions
of what it means for a group or semigroup G  Isom(X) to be discrete. We show

that these notions are equivalent in the Standard Case. Finally, we give examples
to show that these notions are no longer equivalent when X = H∞ .

Throughout this chapter, the standing assumptions that X is a (not necessarly
hyperbolic) metric space and that o ∈ X replace the paper’s overarching standing

assumption that (X, o, b) is a Gromov triple (cf. §4.1). Of course, if (X, o, b) is a
Gromov triple then X is a metric space and o ∈ X, and therefore all theorems in

this chapter can be used in other chapters without comment.

5.1. Topologies on Isom(X)
In this section we discuss different topologies that may be put on the isometry
group of the metric space X.
In the Standard Case, the most natural topology is the compact-open topology
(COT), i.e. the topology whose subbasic open sets are of the form
G(K, U ) = {f ∈ Isom(X) : f (K) ⊆ U }
where K ⊆ X is compact and U ⊆ X is open. When we replace X by a metric

space which is not proper, it is tempting to replace the compact-open topology with
a “bounded-open” topology. However, it is hard to define such a topology in a way
that does not result in pathologies. It turns out that the compact-open topology is
still the “right” topology for many applications in an arbitrary metric space. But
we are getting ahead of ourselves.
Let’s start by considering the case where X is an algebraic hyperbolic space,
i.e., X = H = Hα
F , and figure out what topology or topologies we can put on
Isom(H). Recall from Theorem 2.3.3 that
(5.1.1)

Isom(H) ≡ PO∗ (L; Q) ≡ O∗ (L; Q)/ ∼

where L = HFα+1 , Q is the quadratic form (2.2.1), and T1 ∼ T2 means that [T1 ] =
[T2 ] (in the notation of Section 2.3). Thus Isom(H) is isomorphic to a quotient of
a subspace of L(L), the set of bounded linear maps from L to itself. This indicates
83

84

5. DISCRETENESS

that to define a topology or topologies on Isom(H), it may be best to start from the
functional analysis point of view and look for topologies on L(L). In particular, we
will be interested in the following widely used topologies on L(L):
• The uniform operator topology (UOT) is the topology on L(L) which
comes from looking it as a metric space with the metric
d(T1 , T2 ) = kT1 − T2 k = sup{k(T1 − T2 )xk : x ∈ L, kxk = 1}.
• The strong operator topology (SOT) is the topology on L(L) which comes
from looking at it as a subspace of the product space LL . Note that in
this topology,
Tn −
→T
n

⇔

Tn x −
→ T x ∀x ∈ L.
n

The strong operator topology is weaker than the uniform operator topology.
Remark 5.1.1. There are many other topologies used in functional analysis,
for example the weak operator topology, which we do not consider here.
Starting with either the uniform operator topology or the strong operator topology, we may restrict to the subspace O∗ (L; Q) and then quotient by ∼ to induce

a topology on Isom(H) using the identification (5.1.1). For convenience, we will
also call these induced topologies the uniform operator topology and the strong
operator topology, respectively.
We now return to the general case of a metric space X. Define the Tychonoff
topology to be the topology on Isom(X) inherited from the product topology on
XX.
Proposition 5.1.2.
(i) The Tychonoff topology and the compact-open topology on Isom(X) are
identical.
(ii) If X is an algebraic hyperbolic space, then the strong operator topology
is identical to the Tychonoff topology (and thus also to the compact-open
topology).
Proof.
(i) Since subbasic sets in the Tychonoff topology take the form G({x}, U ), it
is clear that the compact-open topology is at least as fine as the Tychonoff
topology. Conversely, suppose that G(K, U ) is a subbasic open set in the
Tychonoff topology, and fix f ∈ G(K, U ). Let ε = d(f (K), X \ U ) > 0,

5.1. TOPOLOGIES ON Isom(X)

and let (xi )n1 be a set of points in K such that K ⊆
f ∈ U :=

n
\

i=1

G({xi }, Nε/3 (f (K))).1

85

Sn
1

B(xi , ε/3). Then

The set U is open in the Tychonoff topology; we claim that U ⊆ G(K, U ).

Indeed, suppose that fe ∈ U. Then for x ∈ K, fix i with x ∈ B(xi , ε/3);
since fe is an isometry, d(fe(x), f (K)) ≤ d(fe(x), fe(xi )) + d(fe(xi ), f (K)) ≤
2ε/3 < ε. It follows that fe(x) ∈ U ; since x ∈ K was arbitrary, fe ∈
G(K, U ).

(ii) It is clear that the strong operator topology is at least as fine as the
Tychonoff topology. Conversely, suppose that a set U ⊆ Isom(H) is open

in the strong operator topology, and fix [T ] ∈ U. Let T ∈ O∗ (L; Q)
be a representative of [T ]. There exist (vi )n1 in L and ε > 0 such that
for all Te ∈ O∗ (L; Q) satisfying k(Te − T )vi k ≤ ε ∀i, we have [Te] ∈ U.

Let f0 = e0 , and let V = hf0 , v1 , . . . , vn i. Extend {f0 } to an F-basis
{f0 , f1 , . . . , fk } of V with the property that BQ (fj1 , fj2 ) = 0 for all j1 6= j2 .
Without loss of generality, suppose that k ≥ 1. For each i = 1, . . . , n we
P
have vi = j fj ci,j for some ci,j ∈ F, so there exists ε2 > 0 such that for
all Te ∈ O∗ (L; Q) satisfying k(Te − T )fj k ≤ ε2 ∀j and kσTe − σT k ≤ ε2 , we
have [Te] ∈ U.
Let

(5.1.2)

and let




{1}
F=R


IF = {1, i}
F=C,



{1, i, j, k} F = Q

F = {e0 } ∪ {e0 ± (1/2)f1 ℓ : j = 1, . . . , k, ℓ ∈ IF }.
Fix ε3 > 0 small to be determined, and for the remainder of this proof
write A ∼ B if kA − Bk is bounded by a constant which tends to zero as
ε → 0. Let
o
n
V = [Te] ∈ Isom(H) : ∀x ∈ F, ∃yx ∈ [Te]([x]) such that kyx − T xk < ε3 .
For each x ∈ F , we have [x] ∈ H, so the set

{[y] ∈ H : ∃y ∈ [y] such that ky − T xk < ε3 }
is open in the natural topology on H. It follows that V is open in the

Tychonoff topology. Moreover, [T ] ∈ V. To complete the proof we show

1Here and elsewhere N (S) = {x ∈ X : d(x, S) ≤ ε}.
ε

86

5. DISCRETENESS

that V ⊆ U. Indeed, fix [Te] ∈ V, and let y = ye0 . There exists a
representative Te ∈ O∗ (L; Q) such that Tee0 = λy for some λ > 0. Since
−1 = Q(e0 ) ∼ Q(y) = λ−2 Q(λy) = −λ−2 ,

we have λ ∼ 1 and thus Tee0 ∼ T e0 .
Now for each x ∈ F \{e0 }, there exists ax ∈ F such that yx = Te(xax ).
Fix j = 1, . . . , k and ℓ ∈ IF . Writing a± = ae0 ±(1/2)fj ℓ , we have



 
1
1
T e0 ± fj ℓ − Te
< ε3 ,
e0 ± fj ℓ a±
2
2

i.e. T (e0 ± (1/2)fj ℓ) ∼ Te((e0 ± (1/2)fj ℓ)a± ). Substituting ± = + and
± = − and adding the resulting equations gives
1
2T e0 ∼ Te(e0 (a+ + a− )) + Te(fj ℓ(a+ − a− ));
2
using T e0 ∼ Tee0 and rearranging gives

1
Te(e0 (2 − a+ − a− )) ∼ Te(fj ℓ(a+ − a− )).
2
Now by Lemma 2.4.11, we have kTek ∼ 1, and thus e0 (2 − a+ − a− ) ∼
(1/2)fj ℓ(a+ − a− ). Since ke0 a + fj ℓbk ≍× max(|a|, |b|) for all a, b ∈ F, it
follows that

2 − a+ − a− ∼ ℓ(a+ − a− ) ∼ 0,
from which we deduce a+ ∼ a− ∼ 1. Thus




1
1
e
T e0 ± fj ℓ ∼ T e0 ± fj ℓ .
2
2

Substituting ± = + and ± = −, subtracting the resulting equations, and
using the fact that T e0 ∼ Tee0 gives
T (fj ℓ) ∼ Te(fj ℓ).

In particular, letting ℓ = 1 we have T fj ∼ Tefj . Thus
(T fj )(σT ℓ) ∼ (Tefj )(σTe ℓ) ∼ (T fj )(σTe ℓ).

Since this holds for all ℓ ∈ IF , we have σT ∼ σTe . By the definition of ∼,
this means that we can choose ε3 small enough so that kT fj ℓ − Tefj ℓk ≤
ε2 ∀j and kσ e − σT k ≤ ε2 . Then [Te] ∈ U, completing the proof.
T



Proposition 5.1.3. The compact-open topology makes Isom(X) into a topological group, i.e. the maps
(g, h) 7→ gh,

g 7→ g −1

5.2. DISCRETE GROUPS OF ISOMETRIES

87

are continuous.
Proof. Fix g0 , h0 ∈ Isom(X), and let G({x}, U ) be a neighborhood of g0 h0 .

For some ε > 0, we have B(g0 h0 (x), ε) ⊆ U . We claim that

 

G {h0 (x)}, B(g0 h0 (x), ε/2) G {x}, B(h0 (x), ε/2) ⊆ G({x}, U ).

Indeed, fix g ∈ G({h0 (x)}, B(g0 h0 (x), ε/2)) and h ∈ G({x}, B(h0 (x), ε/2)). Then
d(gh(x), g0 h0 (x)) ≤ d(h(x), h0 (x)) + d(gh0 (x), g0 h0 (x)) ≤ ε/2 + ε/2 = ε,
demonstrating that gh(x) ∈ U , and thus that the map (g, h) 7→ gh is continuous.
Now fix g0 ∈ Isom(X), and let G({x}, U ) be a neighborhood of g0−1 . For some
ε > 0, we have B(g0−1 (x), ε) ⊆ U . We claim that
−1
G {g0−1 (x)}, B(x, ε)
⊆ G({x}, U ).
Indeed, fix g ∈ G({g0−1 (x)}, B(x, ε)). Then

d(g −1 (x), g0−1 (x)) = d(x, gg0−1 (x)) ≤ ε,
demonstrating that g −1 (x) ∈ U , and thus that the map g 7→ g −1 is continuous. 
Remark 5.1.4 ([109, 9.B(9), p.60]). If X is a separable complete metric space,
then the group Isom(X) with the compact-open topology is a Polish space.

5.2. Discrete groups of isometries
In this section we discuss several different notions of what it means for a group
G ≤ Isom(X) to be discrete, and then we show that they are equivalent in the

Standard Case. However, each of our notions will be distinct when X = H = Hα
F
for some infinite cardinal α.
Definition 5.2.1. Fix G ≤ Isom(X).

• G is called strongly discrete (SD) if for every bounded set B ⊆ X, we have
#{g ∈ G : g(B) ∩ B 6= } < ∞.
• G is called moderately discrete (MD) if for every x ∈ X, there exists an
open set U ∋ x such that
#{g ∈ G : g(U ) ∩ U 6= } < ∞.
• G is called weakly discrete (WD) if for every x ∈ X, there exists an open
set U ∋ x such that

g(U ) ∩ U 6=  ⇒ g(x) = x.

88

5. DISCRETENESS

Remark 5.2.2. Strongly discrete groups are known in the literature as metrically proper, and moderately discrete groups are known as wandering.
Remark 5.2.3. We may equivalently give the definitions as follows:
• G is strongly discrete (SD) if for every R > 0 and x ∈ X,
(5.2.1)

#{g ∈ G : d(x, g(x)) ≤ R} < ∞.
• G is moderately discrete (MD) if for every x ∈ X, there exists ε > 0 such
that

(5.2.2)

#{g ∈ G : d(x, g(x)) ≤ ε} < ∞.
• G is weakly discrete (WD) if for every x ∈ X, there exists ε > 0 such that

(5.2.3)

G(x) ∩ B(x, ε) = {x}.

As our naming suggests, the condition of strong discreteness is stronger than
the condition of moderate discreteness, which is in turn stronger than the condition
of weak discreteness.
Proposition 5.2.4. Any strongly discrete group is moderately discrete, and
any moderately discrete group is weakly discrete.
Proof. It is clear from the second formulation that strongly discrete groups
are moderately discrete. Let G ≤ Isom(X) be a moderately discrete group. Fix
x ∈ X, and let ε > 0 be such that (5.2.2) holds. Letting ε′ = ε ∧ min{d(x, g(x)) :
g(x) 6= x, g(x) ∈ B(x, ε)}, we see that (5.2.3) holds.


The reverse directions, WD ⇒ MD and MD ⇒ SD, both fail in infinite dimensions. Examples 11.1.14 and 13.3.1-13.3.3 are moderately discrete groups which are
not strongly discrete, and Examples 13.5.2 and 13.4.1 are weakly discrete groups
which are not moderately discrete.
If X is a proper metric space, then the classes MD and SD coincide, but are still
distinct from WD. Example 13.4.1 is a weakly discrete group acting on a proper
metric space which is not moderately discrete. We show now that MD ⇔ SD when
X is proper:
Proposition 5.2.5. Suppose that X is proper. Then a subgroup of Isom(X) is
moderately discrete if and only if it is strongly discrete.
Proof. Let G ≤ Isom(X) be a moderately discrete subgroup. Fix x ∈ X, and
let ε > 0 satisfy (5.2.2). Fix R > 0 and let K = G(o) ∩ B(x, R); K is compact

since X is proper. The collection {B(g(x), ε) : g ∈ G} covers K, so there is a finite

5.2. DISCRETE GROUPS OF ISOMETRIES

89

subcover {B(gi (x), ε) : i = 1, . . . , n}. Now
#{g ∈ G : d(x, g(x) ≤ R)} ≤

n
X
i=1

#{g ∈ G : g(x) ∈ B(gi (x), ε)} < ∞,

i.e. (5.2.1) holds.



5.2.1. Topological discreteness.
Definition 5.2.6. Let T be a topology on Isom(X). A group G ≤ Isom(X)

is T -discrete if it is discrete as a subspace of Isom(X) in the topology T .

Most of the time, we will let T be the compact-open topology (COT). The
relation between COT-discreteness and our previous notions of discreteness is as
follows:
Proposition 5.2.7.
(i) Any moderately discrete group is COT-discrete.
(ii) Any weakly discrete group that is acting on an algebraic hyperbolic space
is COT-discrete.
(iii) Any COT-discrete group that is acting on a proper metric space is strongly
discrete.
Proof.
(i) Let G ≤ Isom(X) be moderately discrete, and let ε > 0 satisfy (5.2.2).
Then the set U := G({o}, B(o, ε)) ⊆ Isom(X) satisfies #(U ∩ G) < ∞.

But U is a neighborhood of id in the compact-open topology. It follows
that G is COT-discrete.

(ii) Suppose that X = H = Hα
F . Let G ≤ Isom(H) be weakly discrete, and by
contradiction suppose it is not COT-discrete. For any finite set F ⊆ H,

let ε > 0 be small enough so that (5.2.3) holds for all x ∈ F ; since G is
not COTD, there exists g = gF ∈ G \ {id} such that d(x, g(x)) ≤ ε for all
x ∈ F , and it follows that g(x) = x for all x ∈ F . Now suppose that J is
a finite set of indices, and let F = {[e0 ]} ∪ {[e0 ± (1/2)ei ]ℓ : i ∈ J, ℓ ∈ IF },

where IF is as in (5.1.2). Then if TI is a representative of gF satisfying
TJ e0 = e0 , an argument similar to the proof of Proposition 5.1.2(ii) shows
that σTJ = I and TJ ei = ei for all i ∈ J.
Now we define an infinite sequence of indices (in )∞
1 as follows: If
i1 , . . . , in−1 have been defined, let Tn = T{i1 ,...,in−1 } , and let in be such
/ Fix(Tn ).
that ein ∈
Choose a nonnegative summable sequence (tn )∞
1 , and let x = e0 +
P∞
n=1 tn ein . Then Tn x → x; since G is weakly discrete, it follows that

90

5. DISCRETENESS

Tn x = x for all n sufficiently large. Fix such an n, and observe that
X
0 = Tn x − x = tn (Tn (en ) − en ) +
tm (Tn (em ) − em );
m>n

the triangle inequality gives
tn ≤

P

2tm
·
kTn en − en k
m>n

By choosing the sequence (tn )∞
1 to satisfy
1
1
kTn en − en ktn ≤ tn ,
4
2
we arrive at a contradiction.
(iii) Let G be a COT-discrete group acting by isometries on a proper metric
tn+1 <

space X. By contradiction, suppose that G is not strongly discrete. Then
there exists an infinite set A ⊆ G such that the set A(o) is bounded.

Without loss of generality we may suppose that A−1 = A. Note that for
each x ∈ X, the set A(x) is bounded and therefore precompact. Now

since X is a proper metric space, it is σ-compact and therefore separable.
Let S be a countable dense subset of X. Then

2
Y
K := 
A(q)
q∈S

is a compact metrizable space. For each g ∈ A let

φg := (g(q))q∈S , (g −1 (q))q∈S ∈ K.

Since A is infinite, there exists an infinite sequence (gn )∞
1 in A such that


φgn → (yq(+) )q∈S , (yq(−) )q∈S ∈ K.
Thus

gn± (q) −
→ yq(±) ∀q ∈ S.
n

The density of S and the equicontinuity of the sequences (gn )∞
1 and
(±)
(±)
±
(gn−1 )∞
imply
that
for
all
x
∈
X,
there
exist
y
such
that
g
(y)
→
yx .
x
1
n

Thus, the sequence (gn )∞
1 converges in the Tychonoff topology to some
(+)
X
(−)
g
∈ X . Similarly, the sequence (gn−1 )∞
∈ XX.
1 converges to some g
−1 ∞
Again, the equicontinuity of the sequences (gn )∞
1 and (gn )1 allows us

to take limits and deduce that
g (+) g (−) = lim gn gn−1 = id.
n→∞

5.2. DISCRETE GROUPS OF ISOMETRIES

91

Similarly, g (−) g (+) = id. Thus g (+) and g (−) are inverses, and in particular
g (+) ∈ Isom(X). Since gn → g (+) in the compact-open topology, the proof
is completed by the following lemma from topological group theory:
Lemma 5.2.8. Let H be a topological group, and let G be a subgroup
of H. Suppose there is a sequence (gn )∞
1 of distinct elements in G which
converges to an element of H. Then G is not discrete in the topology
inherited from H.
Proof. Suppose gn → h ∈ H. Then
−1
gn gn+1
→ hh−1 = id,
−1
while on the other hand gn gn+1
6= id (since the sequence (gn )∞
1 consists

of distinct elements). This demonstrates that G is not discrete in the
inherited topology.
⊳


If X is not an algebraic hyperbolic space, then it is possible for a weakly discrete
group to not be COT-discrete; see Example 13.4.1. Conversely, it is possible for a
COT-discrete group to not be weakly discrete; see Examples 13.4.9 amd 13.5.1.
On the other hand, suppose that X is an algebraic hyperbolic space. The
uniform operator topology (abbreviated as UOT) is finer than the COT, i.e. it has
more open sets, and therefore it is easier for every subset of G to be relatively open
in that topology, which is exactly what it means to be discrete. Notice that there is
an “order switch” here; the UOT is finer than the COT, but the condition of being
COT-discrete is stronger than the condition of being UOT-discrete. We record this
for later use as the following
Observation 5.2.9. Let X be an algebraic hyperbolic space. If a subgroup
G ≤ Isom(X) is COT-discrete, then it is also UOT-discrete.
The inclusion in the previous observation is strict. A significant example of a
group acting on H∞ which is UOT-discrete but not COT-discrete is described in
Example 13.4.2.
The various relations between the distinct shades of discreteness are somewhat
subtle when first discerned. We speculate that it may be fruitful to study such
distinctions with a finer lens. For the reader’s ease, we summarize the relations
between our different notions of discreteness in Table 1 below.
5.2.2. Equivalence in finite dimensions.

92

5. DISCRETENESS

Proposition 5.2.10. Suppose that X is a finite-dimensional Riemannian manifold. Then the notions of strong discreteness, moderate discreteness, weak discreteness, and COT-discreteness agree. If X is an algebraic hyperbolic space, these
notions also agree with the notion of UOT-discreteness.
Proof. By Propositions 5.2.4 and 5.2.7, the conditions of strong discreteness,
moderate discreteness, and COT-discreteness agree and imply weak discreteness.
Conversely, suppose that G ≤ Isom(X) is weakly discrete, and by contradiction

suppose that G is not COT-discrete. Since X is separable, so is Isom(X), and thus
there exists a sequence Isom(X)\{id} ∋ gn → id in the compact-open topology. For
S∞
each n let Fn = {x ∈ X : gn (x) = x}. Since G is weakly discrete, X = 1 Fn , so by
the Baire category theorem, Fn has nonempty interior for some n. But then gn = id
on an open set; in particular there exists a point x0 ∈ X such that gn (x0 ) = x0
and gn′ (x0 ) is the identity map on the tangent space of x0 . By the naturality of the
exponential map, this implies that gn is the identity map, a contradiction.
Finally, suppose X = H = Hα
F is an algebraic hyperbolic space, and let L =

Lα+1
. Since L is finite-dimensional, the SOT and UOT topologies on L(L) are
F
equivalent. This in turn demonstrates that the notions of COT-discreteness and
UOT-discreteness agree.



In such a setting, we shall call a group satisfying any of these equivalent definitions simply discrete.
5.2.3. Proper discontinuity.
Definition 5.2.11. A group G ≤ Isom(X) acts properly discontinuously (PrD)

on X if for every x ∈ X, there exists an open set U ∋ x with
g(U ) ∩ U 6=  ⇒ g = id,
or equivalently, if
d(x, {g(x) : g 6= id}) > 0.

Let us discuss the relations between proper discontinuity and some of our notions of discreteness. We begin by noting that even in finite dimensions, the notion
of proper discontinuity is not the same as the notion of discreteness; instead, a
group acts properly discontinuously if and only if it both discrete and torsion-free.
We also remark that in finite dimensions Selberg’s lemma (see e.g. [8]) can be used
to pass from a discrete group to a finite-index subgroup that acts properly discontinuously. However, it is impossible to do this in infinite dimensions; cf. Example
11.2.18.
Although no notion of discreteness implies proper discontinuity, the reverse is
true for certain types of discreteness. Namely, since #{id} = 1 < ∞, we have:

5.2. DISCRETE GROUPS OF ISOMETRIES

93

Observation 5.2.12. Any group which acts properly discontinuously is moderately discrete.
In particular, by combining with Proposition 5.2.5 we see that if X is proper
then any group which acts properly discontinuously is strongly discrete. This provides a connection between our results, in which strong discreteness is often a
hypothesis, and many results from the literature in which proper discontinuity and
properness are both hypotheses.
Observation 5.2.12 admits the following partial converse, which generalizes the
fact that in finite dimensions every discrete torsion-free group acts properly discontinuously:
Remark 5.2.13. If X is a proper CAT(0) space, then a group acts properly
discontinously if and only if it is moderately discrete and torsion free.
Proof. Suppose that G ≤ Isom(X) acts properly discontinuously. If g ∈

G \ {id} is a torsion element, then by Cartan’s lemma [39, II.2.8(1)], g has a fixed
point. This contradicts G acting properly discontinuously. Thus G is torsion-free.

Conversely, suppose that G ≤ Isom(X) is moderately discrete and torsion-free.
Given x ∈ X, let ε > 0 be as in (5.2.3), and by contradiction suppose that there

exists g 6= id such that d(x, g(x)) < ε. By (5.2.3), g(x) = x. But then by (5.2.2),
the set {g n : n ∈ Z} is finite, i.e. g is a torsion element. This is a contradiction, so

G acts properly discontinuously.



We summarize the relations between our various notions of discreteness, together with proper discontinuity, in the following table:
Finite dimensional
Riemannian manifold

General metric space
Infinite dimensional
algebraic hyperbolic space

Proper metric space

SD ↔ MD
↑
PrD
SD → MD
ր
PrD
SD → MD
ր
PrD
SD ↔ MD
↑
PrD

↔

WD
l
COTD ↔ UOTD
→
WD
ց
COTD
→
WD
↓
COTD → UOTD
↔ COTD
↓
WD

Table 1. The relations between different notions of discreteness.
COTD and UOTD stand for discrete with respect to the compactopen and uniform operator topologies respectively. All implications not listed have counterexamples; see Chapter 13.

94

5. DISCRETENESS

5.2.4. Behavior with respect to restrictions. Fix G ≤ Isom(X), and

suppose Y ⊆ X is a subspace of X preserved by G, i.e. g(Y ) = Y for all g ∈ G.
Then G can be viewed as a group acting on the metric space (Y, d ↿Y ).
Observation 5.2.14.
(i) G is strongly discrete ⇔ G ↿ Y is strongly discrete
(ii) G is moderately discrete ⇒ G ↿ Y is moderately discrete

(iii) G is weakly discrete ⇒ G ↿ Y is weakly discrete
(iv) G is T -discrete ⇐ G ↿ Y is T ↿ Y -discrete

(v) G acts properly discontinuously on X ⇒ G acts properly discontinuously
on Y .

In particular, strong discreteness is the only concept which is “independent of
the space being acted on”. It is thus the most robust of all our definitions.
Note that for the notions of topological discreteness like COTD and UOTD,
the order of implication reverses; restricting to a subspace may cause a group to no
longer be discrete. Example 13.4.9 is an example of this phenomenon.
5.2.5. Countability of discrete groups. In finite dimensions, all discrete
groups are countable. In general, it depends on what type of discreteness you are
considering.
Proposition 5.2.15. Fix G ≤ Isom(X), and suppose that either
(1) G is strongly discrete, or
(2) X is separable and G is COT-discrete.
Then G is countable.
Proof. If G is strongly discrete, then
X
X
#(G) ≤
#{g ∈ G : kgk ≤ n} ≤
#(N) = #(N).
n∈N

n∈N

On the other hand, if X is a separable metric space, then by Remark 5.1.4 Isom(X)
is separable metrizable, so it contains no uncountable discrete subspaces.

Remark 5.2.16. An example of an uncountable UOT-discrete subgroup of
Isom(H∞ ) is given in Example 13.4.2, and an example of an uncountable weakly
discrete group acting on a separable R-tree is given in Example 13.4.1. An example
of an uncountable moderately discrete group acting on a (non-separable) R-tree is
given in Remark 13.3.4.

CHAPTER 6

Classification of isometries and semigroups
In this chapter we classify subsemigroups G  Isom(X) into six categories,
depending on the behavior of the orbit of the basepoint o ∈ X. We start by
classifying individual isometries, although it will turn out that the category into
which an isometry is classified is the same as the category of the cyclic group that

it generates.
We remark that if X is geodesic and G ≤ Isom(X) is a group, then the main

results of this chapter were proven in [88]. Moreover, our terminology is based on
[48, §3.A], where a similar classification was given based on [85, § 3.1].

6.1. Classification of isometries
Fix g ∈ Isom(X), and let
Fix(g) := {x ∈ bord X : g(x) = x}.
Consider ξ ∈ Fix(g) ∩ ∂X. Recall that g ′ (ξ) denotes the dynamical derivative of g

at ξ (see §4.2.3).

Definition 6.1.1. ξ is said to be
• a neutral or indifferent fixed point if g ′ (ξ) = 1,
• an attracting fixed point if g ′ (ξ) < 1, and

• a repelling fixed point if g ′ (ξ) > 1.

Definition 6.1.2. An isometry g ∈ Isom(X) is called
• elliptic if the orbit {g n (o) : n ∈ N} is bounded,

• parabolic if it is not elliptic and has a unique fixed point in ∂X, which is
neutral, and
• loxodromic if it has exactly two fixed points in ∂X, one of which is attracting and the other of which is repelling.

Remark 6.1.3. We use the terminology “loxodromic” rather than the more
common “hyperbolic” to avoid confusion with the many other meanings of the
word “hyperbolic”. In particular, when we get to classification of groups it would
95

96

6. CLASSIFICATION OF ISOMETRIES AND SEMIGROUPS

be a bad idea to call any group “hyperbolic” if it is not hyperbolic in the sense of
Gromov.
The categories of elliptic, parabolic, and loxodromic are clearly mutually exclusive.1 In the converse direction we have the following:
Theorem 6.1.4. Any isometry is either elliptic, parabolic, or loxodromic.
The proof of Theorem 6.1.4 will proceed through several lemmas.
Lemma 6.1.5 (A corollary of [107, Proposition 5.1]). If g ∈ Isom(X) is not
elliptic, then Fix(g) ∩ ∂X 6= .
We include the proof for completeness.
Proof. For each t ∈ N, let nt be the smallest integer such that
kg nt k ≥ nt .
The sequence (nt )∞
1 is nondecreasing. Given s, t ∈ N with s < t, we have
d(g ns (o), g nt (o)) = kg nt −ns k < nt ,
and thus

1
1
[ns + nt − nt ] = ns −→ ∞,
s,t
2
2
nt
nt
i.e. (g (o))t is a Gromov sequence. Let ξ = [(g (o))t ], and note that


hξ|g(ξ)io = lim hg nt (o)|g nt +1 (o)i ≥ lim kg nt k − d(g nt (o), g nt +1 (o)) = ∞.
hg ns (o)|g nt (o)io >

t→∞

t→∞

Thus g(ξ) = ξ, i.e. ξ ∈ Fix(g) ∩ ∂X.



Remark 6.1.6 ([107, Proposition 5.2]). If g ∈ Isom(X) is elliptic and if X is
CAT(0), then Fix(g) ∩ X 6=  due to Cartan’s lemma (Theorem 6.2.5 below). Thus
if X is a CAT(0) space, then any isometry of X has a fixed point in bord X.

Lemma 6.1.7. If g ∈ Isom(X) has an attracting or repelling periodic point,

then g is loxodromic.

Proof. Suppose that ξ ∈ ∂X is a repelling fixed point for g ∈ Isom(X), i.e.
g (ξ) > 1. Recall from Proposition 4.2.8 that
′

Dξ (g n (y1 ), g n (y2 )) ≤ Cg ′ (ξ)−n Dξ (y1 , y2 ) ∀y1 , y2 ∈ Eξ ∀n ∈ Z
for some constant C > 0. Now let n be large enough so that g ′ (ξ)n > C; then
the above inequality shows that the map g n is a strict contraction of the complete
metametric space (Eξ , Dξ ) (cf. Proposition 3.6.19). Then by Theorem 3.6.2, g has
a unique fixed point η ∈ (Eξ )refl = ∂X \ {ξ}. By Corollary 4.2.15, η is an attracting
1Proposition 4.2.16 can be used to show that loxodromic isometries are not elliptic.

6.1. CLASSIFICATION OF ISOMETRIES

97

fixed point. Corollary 4.2.15 also implies that g cannot have a third fixed point.
Thus g is loxodromic.
On the other hand, if g has an attracting fixed point, then by Proposition 4.2.14,
g −1 has a repelling fixed point. Thus g −1 is loxodromic, so applying Proposition
4.2.14 again, we see that g is loxodromic.

Proof of Theorem 6.1.4. By contradiction suppose that g is not elliptic
or loxodromic, and we will show that it is parabolic. By Lemma 6.1.5, we have
Fix(g) ∩ ∂X 6= ; on the other hand, by Lemma 6.1.7, every fixed point of g in
∂X is neutral. It remains to show that #(Fix(g)) = 1. By contradiction, suppose
otherwise. Since g is not elliptic, we clearly have Fix(g) ∩ X = . Thus we may
suppose that there are two distinct neutral fixed points ξ1 , ξ2 ∈ ∂X.
Now for each n ∈ N, we have
B ξi (o, g n (o)) ≍+ n logb (g ′ (ξi )) = 0,

i = 1, 2

by Proposition 4.2.16. Let r = rξ1 ,ξ2 ,o and θ = θξ1 ,ξ2 ,o be as in Section 4.6. Then
by Lemma 4.6.3 we have r(g n (o)) ≍+ θ(g n (o)) ≍+ 0. Thus by Lemma 4.6.2 we
have
kg n k ≍+ |r(g n (o))| + θ(g n (o)) ≍+ 0,

i.e. the sequence {g n (o) : n ∈ N} is bounded. Thus g is elliptic, contradicting our
hypothesis.

Remark 6.1.8 (Cf. [51, Chapter 3, Theorem 1.4]). For R-trees, parabolic
isometries are impossible, so Theorem 6.1.4 shows that every isometry is elliptic or
loxodromic.
Proof. By contradiction suppose that X is an R-tree and that g ∈ Isom(X)

is a parabolic isometry with fixed point ξ ∈ ∂X. Let x = C(o, g(o), ξ) ∈ X; then
x = [o, ξ]t for some t ≥ 0. Now,
d(g(o), x) = kxk + B ξ (g(o), o) = t + 0 = t.
It follows that g(x) = [g(o), ξ]t = x. Thus g is elliptic, a contradiction.



6.1.1. More on loxodromic isometries.
Notation 6.1.9. Suppose g ∈ Isom(X) is loxodromic. Then g+ and g− denote
the attracting and repelling fixed points of g, respectively.
Theorem 6.1.10. Let g ∈ Isom(X) be loxodromic. Then
(6.1.1)

g ′ (g+ ) =

1
·
g ′ (g− )

98

6. CLASSIFICATION OF ISOMETRIES AND SEMIGROUPS

Furthermore, for every x ∈ bord X \ {g− } and for every n ∈ N we have
(6.1.2)

[g ′ (g+ )]n
,
D(g− , g+ )D(x, g− )

D(g n (x), g+ ) .×

with ≤ if X is strongly hyperbolic. In particular
x 6= g− ⇒ g n (x) −
→ g+ ,
n

and the convergence is uniform over any set whose closure does not contain g− .
Finally,
(6.1.3)

kg n k ≍+ |n| logb g ′ (g− ) = |n| logb

1
g ′ (g

+)

·

Proof. (6.1.1) follows directly from Corollary 4.2.15.
To demonstrate (6.1.2), note that
hx|g− io + hg n (x)|g+ io

&+ B g− (o, x) + B g+ (o, g n (x))

(by (j) of Proposition 3.3.3)

≍+ B g− (o, x) + B g+ (o, x) − n logb g ′ (g+ )
′

≍+ hg− |g+ ix − hg− |g+ io − n logb g (g+ )

(by Proposition 4.2.16)
(by (g) of Proposition 3.3.3)

≥ −hg− |g+ io − n logb g ′ (g+ ).

Exponentiating and rearranging yields (6.1.2).
Finally, (6.1.3) follows directly from Lemmas 4.6.2 and 4.6.3.



6.1.2. The story for real hyperbolic spaces. If X is a real hyperbolic
space, then we may conjugate each g ∈ Isom(X) to a “normal form” whose geo-

metrical significance is clearer. The normal form will depend on the classification
of g as elliptic, parabolic, or hyperbolic.
Proposition 6.1.11. Let X be a real hyperbolic space, and fix g ∈ Isom(X).
(i) If g is elliptic, then g is conjugate to a map of the form T ↿ B for some
linear isometry T ∈ O∗ (H).
(ii) If g is parabolic, then g is conjugate to a map of the form x 7→ Tbx + p :

E → E, where T ∈ O∗ (B) and p ∈ B. Here B = ∂E \ {∞} = Hα−1 .
(iii) If g is hyperbolic, then g is conjugate to a map of the form x 7→ λTbx :

E → E, where 0 < λ < 1 and T ∈ O∗ (B).
Proof.

(i) If g is elliptic, then by Cartan’s lemma (Theorem 6.2.5 below), g has a
fixed point x ∈ X. Since Isom(X) acts transitively on X (Observation
2.3.2), we may conjugate to B in a way such that g(0) = 0. But then by
Proposition 2.5.4, g is of the form (i).

6.2. CLASSIFICATION OF SEMIGROUPS

99

(ii) Let ξ be the neutral fixed point of g. Since Isom(X) acts transitively on
∂X (Proposition 2.5.9), we may conjugate to E in a way such g(∞) = ∞.
Then by Proposition 2.5.8 and Example 4.2.11, g is of the form (ii).
(iii) Since Isom(X) acts doubly transitively on ∂X (Proposition 2.5.9), we
may conjugate to E in a way such that g+ = 0 and g− = ∞. Then by

Proposition 2.5.8 and Example 4.2.11, g is of the form (iii). (We have
p = 0 since 0 ∈ Fix(g).)


Remark 6.1.12. If g ∈ Isom(X) is elliptic or loxodromic, then the orbit
(g (o))∞
1 exhibits some “regularity” - either it remains bounded forever, or it din

verges to the boundary. On the other hand, if g is parabolic then the orbit can
oscillate, both accumulating at infinity and returning infinitely often to a bounded
region. This is in sharp contrast to finite dimensions, where such behavior is impossible. We discuss such examples in detail in §11.1.2.
6.2. Classification of semigroups
Notation 6.2.1. We denote the set of global fixed points of a semigroup G 
Isom(X) by
\
Fix(G) :=
Fix(g).
g∈G

Definition 6.2.2. G is
• elliptic if G(o) is a bounded set.
• parabolic if G is not elliptic and has a global fixed point ξ ∈ Fix(G) such
that

g ′ (ξ) = 1 ∀g ∈ G,
i.e. ξ is neutral with respect to every element of G.
• loxodromic if it contains a loxodromic isometry.
Below we shall prove the following theorem:
Theorem 6.2.3. Every semigroup of isometries of a hyperbolic metric space is
either elliptic, parabolic, or loxodromic.
Observation 6.2.4. An isometry g is elliptic, parabolic, or loxodromic according to whether the cyclic group generated by it is elliptic, parabolic, or loxodromic.
A similar statement holds if “group” is replaced by “semigroup”. Thus, Theorem
6.1.4 is a special case of Theorem 6.2.3.
Before proving Theorem 6.2.3, let us say a bit about each of the different
categories in this classification.

100

6. CLASSIFICATION OF ISOMETRIES AND SEMIGROUPS

6.2.1. Elliptic semigroups. Elliptic semigroups are the least interesting of
the semigroups we consider. Indeed, we observe that any strongly discrete elliptic semigroup is finite. We now consider the question of whether every elliptic
semigroup has a global fixed point.
Theorem 6.2.5 (Cartan’s lemma). If X is a CAT(0) space (and in particular
if X is a CAT(-1) space), then every elliptic subsemigroup G  Isom(X) has a
global fixed point.

We remark that if G is a group, then this result may be found as [39, Corollary
II.2.8(1)].
Proof. Since G(o) is a bounded set, it has a unique circumcenter [39, Proposition II.2.7], i.e. the minimum
min sup d(x, g(o))

x∈X g∈G

is achieved at a single point x ∈ X. We claim that x is a global fixed point of G.
Indeed, for each h ∈ G we have
sup d(h−1 (x), g(o)) = sup d(x, hg(o)) ≤ sup d(x, g(o));

g∈G

g∈G

since x is the circumcenter we deduce that h

g∈G

−1

(x) = x, or equivalently that h(x) =

x.



On the other hand, if we do not restrict to CAT(0) spaces, then it is possible
to have an elliptic group with no global fixed point. We have the following simple
example:
Example 6.2.6. Let X = B \ BB (0, 1) and let g(x) = −x. Then X is a
hyperbolic metric space, g is an isometry of X, and G = {id, g} is an elliptic group

with no global fixed point.

6.2.2. Parabolic semigroups. Parabolic semigroups will be important in
Chapter 12 when we consider geometrically finite semigroups. In particular, we
make the following definition:
Definition 6.2.7. Let G  Isom(X). A point ξ ∈ ∂X is a parabolic fixed point

of G if the semigroup

Gξ := Stab(G; ξ) = {g ∈ G : g(ξ) = ξ}
is a parabolic semigroup.
In particular, if G is a parabolic semigroup then the unique global fixed point
of G is a parabolic fixed point.

6.2. CLASSIFICATION OF SEMIGROUPS

101

Warning 6.2.8. A parabolic group does not necessarily contain a parabolic
isometry; see Example 11.2.18.
Note that Proposition 4.2.8 yields the following observation:
Observation 6.2.9. Let G  Isom(X), and let ξ be a parabolic fixed point of

G. Then the action of Gξ on (Eξ , Dξ ) is uniformly Lipschitz, i.e.

Dξ (g(y1 ), g(y2 )) ≍× Dξ (y1 , y2 ) ∀y1 , y2 ∈ Eξ ∀g ∈ G,
and the implied constant is independent of g ∈ G. Furthermore, if X is strongly
hyperbolic, then G acts isometrically on Eξ .
Observation 6.2.10. Let G  Isom(X), and let ξ be a parabolic fixed point
of G. Then for all g ∈ Gξ ,
Dξ (o, g(o)) ≍× b(1/2)kgk ,

(6.2.1)

with equality if X is strongly hyperbolic.
Proof. This is a direct consequence of (3.6.6), (h) of Proposition 3.3.3, and
Proposition 4.2.16.



As a corollary we have the following:
Observation 6.2.11. Let G  Isom(X), and let ξ be a parabolic fixed point
of G. Then for any sequence (gn )∞
1 in Gξ ,
kgn k −
→ ∞ ⇔ gn (o) −
→ ξ.
n

n

Proof. Indeed,
gn (o) −
→ ξ ⇔ Dξ (o, gn (o)) −
→ ∞ ⇔ kgn k −
→ ∞.
n

n

n


Remark 6.2.12. If X is an R-tree, then any parabolic group must be infinitely
generated. This follows from a straightforward modification of the proof of Remark
6.1.8.
6.2.3. Loxodromic semigroups. We now come to loxodromic semigroups,
which are the most diverse out of these classes. In fact, they are so diverse that we
separate them into three subclasses.
Definition 6.2.13 ([48]). Let G  Isom(X) be a loxodromic semigroup. G is
• lineal if Fix(g) = Fix(h) for all loxodromic g, h ∈ G.

• of general type if it has two loxodromic elements g, h ∈ G with
Fix(g) ∩ Fix(h) = .

102

6. CLASSIFICATION OF ISOMETRIES AND SEMIGROUPS

• focal if #(Fix(G)) = 1.
(We remark that focal groups were called quasiparabolic by Gromov [85, §3, Case
4’].
We observe that any cyclic loxodromic group or semigroup is lineal, so this
refined classification does not give any additional information for individual isometries.
Proposition 6.2.14. Any loxodromic semigroup is either lineal, focal, or of
general type.
Proof. Clearly, #(Fix(G)) ≤ 2 for any loxodromic semigroup G; moreover,
#(Fix(G)) = 2 if and only if G is lineal. So to complete the proof, it suffices
to show that #(Fix(G)) = 0 if and only if G is of general type. The backward
direction is obvious. Suppose that #(Fix(G)) = 0, but that G is not of general type.
Combinatorial considerations show that there exist three points ξ1 , ξ2 , ξ3 ∈ ∂X such
that Fix(g) ⊆ {ξ1 , ξ2 , ξ3 } for all g ∈ G. But then the set {ξ1 , ξ2 , ξ3 } would have to

be preserved by every element of g, which contradicts the definition of a loxodromic
isometry.

Let G be a focal semigroup, and let ξ be the global fixed point of G. The
dynamics of G will be different depending on whether or not g ′ (ξ) > 1 for any

g ∈ G.
Definition 6.2.15. G will be called outward focal if g ′ (ξ) > 1 for some g ∈ G,

and inward focal otherwise.

Note that an inward focal semigroup cannot be a group.
Proposition 6.2.16. For G ≤ Isom(X), the following are equivalent:
(A) G is focal.
(B) G has a unique global fixed point ξ ∈ ∂X, and g ′ (ξ) 6= 1 for some g ∈ G.

(C) G has a unique global fixed pont ξ ∈ ∂X, and there are two loxodromic
isometries g, h ∈ G so that g+ = h+ = ξ, but g− 6= h− .

Proof. The implications (C) ⇒ (A) ⇔ (B) are straightforward. Suppose that
G is focal, and let g ∈ G be a loxodromic isometry. Since G is a group, we may

without loss of generality suppose that g ′ (ξ) < 1, so that g+ = ξ. Let j ∈ G be
such that g− ∈
/ Fix(j). By choosing n sufficiently large, we may guarantee that

(jg n )′ (ξ) < 1. Then if h = jg n , then h is loxodromic and h+ = ξ. But g− ∈
/ Fix(h),
so g− 6= h− .


6.3. PROOF OF THE CLASSIFICATION THEOREM

103

6.3. Proof of the Classification Theorem
We begin by recalling the following definition from Section 4.5:
Definition 4.5.1. For each σ > 0 and x, y ∈ X, let
Shady (x, σ) = {η ∈ ∂X : hy|ηix ≤ σ}.
We say that Shady (x, σ) is the shadow cast by x from the light source y, with
parameter σ. For shorthand we will write Shad(x, σ) = Shado (x, σ).
Lemma 6.3.1. For every σ > 0, there exists r > 0 such that for every g ∈
Isom(X) with kgk ≥ r, if there exists a nonempty closed set
Z ⊆ Shadg−1 (o) (o, σ)
satisfying g(Z) ⊆ Z, then g is loxodromic and g+ ∈ Z.
Proof. Recall from the Bounded Distortion Lemma 4.5.6 that
(6.3.1)

D(g(y1 ), g(y2 ))
≤ Cb−kgk ∀y1 , y2 ∈ Z
D(y1 , y2 )

for some C > 0 independent of g. Now choose r > 0 large enough so that Cb−r < 1.
If g ∈ Isom(X) satisfies kgk ≥ r, we can conclude that g : Z → Z is a strict
contraction of the complete metametric space (Z, D). Then by Theorem 3.6.2, g
has a unique fixed point ξ ∈ Zrefl = Z ∩ ∂X.

To complete the proof we must show that g ′ (ξ) < 1 to prove that g is not
parabolic and that ξ = g+ . Indeed, by the Bounded Distortion Lemma, we have
g ′ (ξ) .× b−kgk ≤ b−r , so choosing r sufficiently large completes the proof.



Corollary 6.3.2. For every σ > 0, there exists r = rσ > 0 such that for every
g ∈ Isom(X) with kgk ≥ r, if g is not loxodromic, then
hg(o)|g −1 (o)io ≥ σ.

(6.3.2)

Proof. Fix σ > 0, and let σ ′ = σ + δ, where δ is the implied constant in
Gromov’s inequality. Apply Lemma 6.3.1 to get r′ > 0. Let r = max(r′ , 2σ ′ ). Now
suppose that g ∈ Isom(X) satisfies kgk ≥ r ≥ r′ but is not loxodromic. Then by
Lemma 6.3.1, we have
Shad(g(o), σ ′ ) \ Shadg−1 (o) (o, σ ′ ) 6= .
Let x be a member of this set. By definition this means that
ho|xig(o) ≤ σ ′ < hg −1 (o)|xio .
Since kgk ≥ r ≥ 2σ ′ , we have
hg(o)|xio = kgk − ho|xig(o) ≥ 2σ ′ − σ ′ = σ ′ .

104

6. CLASSIFICATION OF ISOMETRIES AND SEMIGROUPS

Now by Gromov’s inequality we have
hg(o)|g −1 (o)io ≥ min(hg(o)|xio , hg −1 (o)|xio ) − δ ≥ σ ′ − δ = σ.

Lemma 6.3.3. Let G  Isom(X) be a semigroup which is not loxodromic, and
∞
let (gn )∞
1 be a sequence in G such that kgn k → ∞. Then (gn (o))1 is a Gromov
sequence.

Proof. Fix σ > 0 large, and let r = rσ be as in Corollary 6.3.2. Since G is
not loxodromic, (6.3.2) holds for every g ∈ G for which kgk ≥ r.
Fix n, m ∈ N with kgn k, kgm k ≥ r; Corollary 6.3.2 gives
hgn (o)|gn−1 (o)io ≥ σ

(6.3.3)

−1
hgm (o)|gm
(o)io ≥ σ.

(6.3.4)

By contradiction, suppose that hgn (o)|gm (o)io ≤ σ/2; then Gromov’s inequality
together with (6.3.3) gives
hgn−1 (o)|gm (o)io ≍+ 0.

(6.3.5)
It follows that

kgn gm k = d(gn−1 (o), gm (o)) ≥ 2r − hgn−1 (o)|gm (o)io ≍+ 2r.
Choosing r sufficiently large, we have kgn gm k ≥ r. So by Corollary 6.3.2,
(6.3.6)

−1 −1
hgn gm (o)|gm
gn (o)io ≥ σ.

Now
hgn (o)|gn gm (o)io = ho|gm (o)ign−1 (o)

= kgn k − hgn−1 (o)|gm (o)io
≍+ kgn k

(by (6.3.5))

≥ r,
i.e.
(6.3.7)

hgn (o)|gn gm (o)io &+ r.

A similar argument yields
(6.3.8)

−1
−1 −1
hgm
(o)|gm
gn (o)io &+ r.

Combining (6.3.4), (6.3.8), (6.3.6), and (6.3.7), together with Gromov’s inequality,
yields
hgn |gm io &+ min(σ, r).

6.4. DISCRETENESS AND FOCAL GROUPS

This completes the proof.

105



Proof of Theorem 6.2.3. Suppose that G is neither elliptic nor loxodromic,
and we will show that it is parabolic. Since G is not elliptic, there is a sequence
∞
(gn )∞
1 in G such that kgn k → ∞. By Lemma 6.3.3, (gn (o))1 is a Gromov sequence;

let ξ ∈ ∂X be the limit point.
Note that ξ is uniquely determined by G; if (hn (o))∞
1 were another Gromov

sequence, then we could let
jn :=


g

n/2

h

(n−1)/2

n even

.

n odd

The sequence (jn (o))∞
1 would tend to infinity, so by Lemma 6.3.3 it would be a
Gromov sequence. But that exactly means that the Gromov sequences (gn (o))∞
1
and (hn (o))∞
1 are equivalent. Moreover, it is easy to see that ξ does not depend on
the choice of the basepoint o ∈ X.
In particular, the fact that ξ is canonically determined by G implies that ξ is

a global fixed point of G. To complete the proof, we need to show that g ′ (ξ) = 1
for all g ∈ G. Suppose we have g ∈ G such that g ′ (ξ) 6= 1. Then g is loxodromic

by Lemma 6.1.7, a contradiction.



6.4. Discreteness and focal groups
Proposition 6.4.1. Fix G ≤ Isom(X), and suppose that either
(1) G is strongly discrete,
(2) X is CAT(-1) and G is moderately discrete, or
(3) X admits unique geodesic extensions (e.g. X is an algebraic hyperbolic
space) and G is weakly discrete.
Then G is not focal.
Strongly discrete case. Suppose that G is a focal group. Let ξ ∈ ∂X be
its global fixed point, and let g, h ∈ G be as in (C) of Proposition 6.2.16. Since
h−n (o) → h− 6= ξ, we have

hh−n (o)|ξio ≍+,h 0

and thus
hhn (o)|ξio ≍+ khn k − ho|ξihn (o) ≍+,h khn k.
Applying g we have
hghn (o)|ξio = hhn (o)|ξig−1 (o) ≍+,g hhn (o)|ξio ≍+,h khn k

106

6. CLASSIFICATION OF ISOMETRIES AND SEMIGROUPS

and applying Gromov’s inequality we have
hhn (o)|ghn (o)io ≍+,g,h khn k ≍+,g kghn k.
Now
kh−n ghn k = d(hn (o), ghn (o))

= khn k + kghn k − 2hhn (o), ghn (o)io ≍+,g,h 0.

Since G is strongly discrete, this implies that the collection {h−n ghn : n ∈ N} is
finite, and so for some n1 < n2 we have
h−n1 ghn1 = h−n2 ghn2
or
hn2 −n1 g = ghn2 −n1 ,
i.e. hn2 −n1 commutes with g. But then hn2 −n1 (g− ) = g− , contradicting that
g− 6= h− . This completes the proof of Proposition 6.4.1(1).

Moderately discrete case. Suppose that G is a focal group. Let ξ ∈ ∂X
be its global fixed point, and let g, h ∈ G be as in (C) of Proposition 6.2.16. Let
k = [g, h] = g −1 h−1 gh ∈ G.
We observe first that
(6.4.1)

k ′ (ξ) =

1 ′
1
g (ξ)h′ (ξ) = 1.
g ′ (ξ) h′ (ξ)

Note that strong hyperbolicity is necessary to deduce equality in (6.4.1) rather than
merely a coarse asymptotic.
Next, we claim that k(g− ) 6= g− . Indeed, g− ∈
/ Fix(h), so h(g− ) 6= g− . This

in turn implies that h(g− ) ∈
/ Fix(g), so gh(g− ) 6= h(g− ). Now applying g −1 h−1 to
both sides shows that k(g− ) 6= g− .
Claim 6.4.2. g −n kg n (o) → o.
Proof. Indeed,
kg −n kg n k = d(g n (o), kg n (o)).
Let
x=ξ
y=o
z = k(o)
pn = g n (o)
qn = kg n (o).

6.4. DISCRETENESS AND FOCAL GROUPS

107

x=ξ

pn = gn (o)

o

qn = kgn (o)

z = k(o)

Figure 6.4.1. The higher the point g n (o) is, the smaller its displacement under k is.

(See Figure 6.4.1.) Then pn , qn ∈ ∆ := ∆(x, y, z). By Proposition 4.4.13 d(pn , qn ) ≤

d(pn , q n ), where pn , q n are comparison points for pn , qn on the comparison triangle
∆ = ∆(x, y, z). Now notice that
B x (pn , q n ) = B ξ (g n (o), kg n (o)) = 0
by Proposition 4.2.16 and (6.4.1). On the other hand, pn , q n → x. An easy

calculation based on (2.5.3) and Proposition 3.5.5 (letting x = ∞) shows that
d(pn , q n ) → 0, and thus that kg −n kg n k → 0 i.e. g −n kg n (o) → o.

⊳

Since G is moderately discrete, this implies that the collection {g −n kg n : n ∈
N} is finite. As before (in the proof of the strongly discrete case), this implies
that g n and k commute for some n ∈ N. But (g n )− = g− , and k(g− ) 6= g− ,
which contradicts that g n and k commute. This completes the proof of Proposition
6.4.1(2).



Weakly discrete case. Suppose that G is a focal group. Let ξ, g, h, and k
be as above. Without loss of generality, supposet that o ∈ [g− , ξ].
Claim 6.4.3. g −n kg n (o) 6= o for all n ∈ N.

108

6. CLASSIFICATION OF ISOMETRIES AND SEMIGROUPS

Proof. Fix n ∈ N. As observed above, k(g− ) 6= g− . On the other hand,

k(ξ) = ξ, and g n (o) ∈ [g− , ξ]. Since X admits unique geodesic extensions, it follows
that k(g n (o)) 6= g n (o), or equivalently that g −n kg n (o) 6= o.
⊳
Together with Claim 6.4.2, this contradicts that G is weakly discrete. This
completes the proof of Proposition 6.4.1(3).


CHAPTER 7

Limit sets
Throughout this chapter, we fix a subsemigroup G  Isom(X). We define
the limit set of G, along with various subsets. We then define several concepts in
terms of the limit set including elementariness and compact type, while relating
other concepts to the limit set, such as the quasiconvex core and irreducibility of a
group action. We also prove that the limit set is minimal in an approprate sense
(Proposition 7.4.1 - Proposition 7.4.6).
7.1. Modes of convergence to the boundary
We recall (Observation 3.4.20) that a sequence (xn )∞
1 in X converges to a point
ξ ∈ ∂X if and only if

hxn |ξio −
→ ∞.
n

In this section we define more restricted modes of convergence. To get an intuition
let us consider the case where X = E = Eα is the half-space model of a real
hyperbolic space. Consider a sequence (xn )∞
1 in E which converges to a point
α−1
ξ ∈ B := ∂E \ {∞} = H
. We say that xn → ξ conically if there exists θ > 0
such that if we let

C(ξ, θ) = {x ∈ E : x1 ≥ sin(θ)kx − ξk}
then xn ∈ C(ξ, θ) for all n ∈ N. We call C(ξ, θ) the cone centered at ξ with
inclination θ; see Figure 7.1.1.

Proposition 7.1.1. Let (xn )∞
1 be a sequence in E converging to a point ξ ∈ B.

Then the following are equivalent:

(A) (xn )∞
1 converges conically to ξ.
(B) The sequence (xn )∞
1 lies within a bounded distance of the geodesic ray
[o, ξ].
(C) There exists σ > 0 such that for all n ∈ N,
ho|ξixn ≤ σ,
or equivalently
(7.1.1)

ξ ∈ Shad(xn , σ).
109

110

7. LIMIT SETS

C(ξ, θ)

θ
ξ

Figure 7.1.1. A sequence converging conically to ξ. For each
point x, the height of x is greater than sin(θ) times the distance
from x to ξ.

Moreover, the equivalence of (B) and (C) holds in all geodesic hyperbolic metric
spaces.
Proof. The equivalence of (B) and (C) follows directly from (i) of Proposition
4.3.1. Moreover, conditions (B) and (C) are clearly independent of the basepoint o.
Thus, in proving (A) ⇔ (B) we may without loss of generality suppose that ξ = 0
and o = (1, 0). Note that if θ > 0 is fixed, then
C(0, θ) = {x ∈ E : ∡(x) ≤ π/2 − θ} = {x ∈ E : θ(x) ≤ − log cos(π/2 − θ)},
where θ = θ0,∞,o is as in Proposition 4.6.4. Since − log cos(π/2 − θ) → ∞ as θ → 0,
we have (A) if and only if the sequence (θ(xn ))∞
1 is bounded. But
θ(xn ) = h0|∞ixn ≍+ d(xn , [0, ∞])

(by (i) of Proposition 4.3.1)

= d(xn , [o, 0]),

(for n sufficiently large)

which completes the proof.



Condition (B) of Proposition 7.1.1 motivates calling this kind of convergence
radial ; we shall use this terminology henceforth. However, condition (C) is best
suited to a general hyperbolic metric space.
Definition 7.1.2. Let (xn )∞
1 be a sequence in X converging to a point ξ ∈ ∂X.

We will say that (xn )∞
1 converges to ξ

• σ-radially if (7.1.1) holds for all n ∈ N,

• radially if it converges σ-radially for some σ > 0,
• σ-uniformly radially if it converges σ-radially, x1 = o, and
d(xn , xn+1 ) ≤ σ ∀n ∈ N,
• uniformly radially if it converges σ-uniformly radially for some σ > 0.

7.1. MODES OF CONVERGENCE TO THE BOUNDARY

111

ξ

Figure 7.1.2. A sequence converging horospherically but not radially to ξ.
Note that a sequence can converge σ-radially and uniformly radially without
converging σ-uniformly radially.
We next define horospherical convergence. Again, we motivate the discussion
by considering the case of a real hyperbolic space X = E = Eα . This time, however,
we will let ξ = ∞, and we will say that a sequence (xn )∞
1 converges horospherically
to ξ if

height(xn ) −
→ ∞,
n

where the height of a point x ∈ E is its first coordinate x1 . This terminology
comes from defining a horoball centered at ∞ to be a set of the form H∞,t = {x :

height(x) > et }; then xn → ∞ horospherically if and only if for every horoball
H∞,t centered at infinity, we have xn ∈ H∞,t for all sufficiently large n. (See also
Definition 12.1.1 below.)
Recalling (cf. Proposition 3.5.5) that

height(x) = bB∞ (o,x) ,
the above discussion motivates the following definition:
Definition 7.1.3. A sequence (xn )∞
1 in X converges horospherically to a point
to ξ ∈ ∂X if

B ξ (o, xn ) −
→ +∞.
n

Observation 7.1.4. If xn → ξ radially, then xn → ξ horospherically.
Proof. Indeed,
→ ∞.
B ξ (o, xn ) ≍+ kxn k − 2ho|ξixn ≍+ kxn k −
n


The converse is false, as illustrated in Figure 7.1.2.

112

7. LIMIT SETS

Observation 7.1.5. The concepts of convergence, radial convergence, uniformly radial convergence, and horospherical convergence are independent of the
basepoint o, whereas the concepts of σ-radial convergence and σ-uniformly radial
convergence depend on the basepoint. (Regarding σ-radial convergence, this dependence on basepoint is not too severe; see Proposition 7.2.3 below.)

7.2. Limit sets
We define the limit set of G, a subset of ∂X which encodes geometric information about G. We also define a few important subsets of the limit set.
Definition 7.2.1. Let
N
Λ(G) := {η ∈ ∂X : gn (o) → η for some (gn )∞
1 ∈ G }

N
Λr (G) := {η ∈ ∂X : gn (o) → η radially for some (gn )∞
1 ∈G }

N
Λr,σ (G) := {η ∈ ∂X : gn (o) → η σ-radially for some (gn )∞
1 ∈G }

N
Λur (G) := {η ∈ ∂X : gn (o) → η uniformly radially for some (gn )∞
1 ∈G }

N
Λur,σ (G) := {η ∈ ∂X : gn (o) → η σ-uniformly radially for some (gn )∞
1 ∈ G }
N
Λh (G) := {η ∈ ∂X : gn (o) → η horospherically for some (gn )∞
1 ∈ G }.

These sets are respectively called the limit set, radial limit set, σ-radial limit set,
uniformly radial limit set, σ-uniformly radial limit set, and horospherical limit set
of the semigroup G.
Note that
Λr =

[

Λr,σ

σ>0

Λur =

[

Λur,σ

σ>0

Λur ⊆ Λr ⊆ Λh ⊆ Λ.
Observation 7.2.2. The sets Λ, Λr , Λur , and Λh are invariant1 under the
action of G, and are independent of the basepoint o. The set Λ is closed.
Proof. The first assertion follows from Observation 7.1.5 and the second follows directly from the definition of Λ as the intersection of ∂X with the set of
accumulation points of the set G(o).

1By invariant we always mean forward invariant.

7.3. CARDINALITY OF THE LIMIT SET

113

Proposition 7.2.3 (Near-invariance of the sets Λr,σ ). For every σ > 0, there
exists τ > 0 such that for every g ∈ G, we have
g(Λr,σ ) ⊆ Λr,τ .

(7.2.1)

If X is strongly hyperbolic, then (7.2.1) holds for all τ > σ.
Proof. Fix ξ ∈ Λr,σ . There exists a sequence (hn )∞
1 so that hn (o) → ξ
σ-radially, i.e.
ho|ξihn (o) ≤ σ ∀n ∈ N
and hn (o) → ξ. Now
ho|g −1 (o)ihn (o) ≥ khn k − kg −1 k −
→ +∞.
n

Thus, for n sufficiently large, Gromov’s inequality gives
hg −1 (o)|ξihn (o) .+ σ

(7.2.2)
i.e.

ho|g(ξ)ighn (o) .+ σ.
So ghn (o) → g(ξ) τ -radially, where τ is equal to σ plus the implied constant of this
asymptotic. Thus, g(ξ) ∈ Λr,τ .

If X is strongly hyperbolic, then by using (3.3.6) instead of Gromov’s inequality,
the implied constant of (7.2.2) can be made arbitrarily small. Thus τ may be taken
arbitrarily close to σ.



7.3. Cardinality of the limit set
In this section we characterize the cardinality of the limit set according to the
classification of the semigroup G.
Proposition 7.3.1 (Cardinality of the limit set by classification). Fix G 
Isom(X).
(i) If G is elliptic, then Λ = .
(ii) If G is parabolic or inward focal with global fixed point ξ, then Λ = {ξ}.
(iii) If G is lineal with fixed pair {ξ1 , ξ2 }, then Λ ⊆ {ξ1 , ξ2 }, with equality if G
is a group.

(iv) If G is outward focal or of general type, then #(Λ) ≥ #(R). Equality holds
if X is separable.
Case (i) is immediate, while case (iv) requires the theory of Schottky groups
and will be proven in Chapter 10 (see Proposition 10.5.4).

114

7. LIMIT SETS

Proof of (ii). For g ∈ G, g ′ (ξ) ≤ 1, so by Proposition 4.2.16, we have

B ξ (g(o), o) .+ 0. In particular, by (h) of Proposition 3.3.3 we have
hx|ξio &+

1
kxk ∀x ∈ G(o).
2

This implies that xn → ξ for any sequence (xn )∞
1 in G(o) satisfying kxn k → ∞. It
follows that Λ = {ξ}.

Proof of (iii). By Lemma 4.6.3 we have
θ(g(o)) ≍+ θ(o) = o ∀g ∈ G,
where θ = θξ1 ,ξ2 ,o = θξ2 ,ξ1 ,o is as in Section 4.6. Thus
hξ1 |ξ2 ix ≍+ 0 ∀x ∈ G(o).
Fix a sequence G(o) ∋ xn → ξ ∈ Λ. By Gromov’s inequality, there exists i = 1, 2
such that
ho|ξi ixn ≍+ 0 for infinitely many n.
It follows that xn → ξi radially along some subsequence, and in particular ξ = ξi .
Thus Λ ⊆ {ξ1 , ξ2 }.

Definition 7.3.2. Fix G  Isom(X). G is called elementary if #(Λ) < ∞ and
nonelementary if #(Λ) = ∞.
Thus, according to Proposition 7.3.1, elliptic, parabolic, lineal, and inward
focal semigroups are elementary while outward focal semigroups and semigroups of
general type are nonelementary.
Remark 7.3.3. In the Standard Case, some authors (e.g. [148, §5.5]) define a
subgroup of Isom(X) to be elementary if there is a global fixed point or a global fixed
geodesic line. According to this definition, focal groups are considered elementary.
By contrast, we follow [48] and others in considering them to be nonelementary.
Another common definition in the Standard Case is that a group is elementary
if it is virtually abelian. This agrees with our definition, but beyond the Standard
Case this equivalence no longer holds (cf. Observation 11.1.4 and Remark 11.1.6).

7.4. Minimality of the limit set
Observation 7.2.2 identified the limit set Λ as a closed G-invariant subset of the
Gromov boundary ∂X. In this section, we give a characterization of Λ depending
on the classification of G.
Proposition 7.4.1 (Cf. [53, Théorème 5.1]). Fix G  Isom(X). Then any

closed G-invariant subset of ∂X containing at least two points contains Λ.

7.4. MINIMALITY OF THE LIMIT SET

115

Proof. We begin with the following lemma, which will also be useful later:
(1)

(2)

∞
∞
Lemma 7.4.2. Let (xn )∞
1 , (yn )1 , (yn )1 be sequences in bord X satisfying

hyn(1) |yn(2) ixn ≍+ 0
and
(i)

xn → ξ ∈ ∂X.

Then ξ ∈ {yn : n ∈ N, i = 1, 2}.
Proof. For n ∈ N fixed, by Gromov’s inequality there exists in = 1, 2 such
that
ho|yn(in ) ixn ≍+ 0.
It follows that
hxn |yn(in ) io ≍+ kxn k −
→ ∞.
n

On the other hand
hxn |ξio −
→ ∞,
n

so by Gromov’s inequality
hyn(in ) |ξio −
→ ∞,
n

i.e.

(i )
yn n

→ ξ.

⊳

Now let F be a closed G-invariant subset of ∂X containing two points ξ1 6= ξ2 ,
and let η ∈ Λ. Then there exists a sequence (gn )∞
1 so that gn (o) → η. Applying
(i)

Lemma 7.4.2 with xn = gn (o) and yn = gn (ξi ) ∈ F completes the proof.



The proof of Proposition 7.4.1 may be compared to the proof of [73, Theorem
3.1], where a quantitative convergence result is proven assuming that η is in the
radial limit set (and assuming that G is a group).
Corollary 7.4.3. Let G  Isom(X) be nonelementary.
(i) If G is outward focal with global fixed point ξ, then Λ is the smallest closed
G-invariant subset of ∂X which contains a point other than ξ.
(ii) (Cf. [20, Theorem 5.3.7]) If G is of general type, then Λ is the smallest
nonempty closed G-invariant subset of ∂X.
Proof. Any G-invariant set containing a point which is not fixed by G contains
two points.



Corollary 7.4.4. Let G  Isom(X) be nonelementary. Then
Λ = Λr = Λur .

116

7. LIMIT SETS

Proof. The implications ⊇ are clear. On the other hand, for each loxodromic

g ∈ G we have g+ ∈ Λur . Thus Λur 6= , and Λur * {ξ} if G is outward focal with
global fixed point ξ. By Proposition 7.3.1, G is either outward focal or of general
type. Applying Corollary 7.4.3, we have Λur ⊇ Λ.



Remark 7.4.5. If G is elementary, it is easily verified that Λ = Λr = Λur unless
G is parabolic, in which case Λr = Λur =  $ Λ.
If G is a nonelementary group, then Corollary 7.4.3 immediately implies that
the set of loxodromic fixed points of G is dense in Λ. However, if G is not a group
then this conclusion does not follow, since the set of attracting loxodromic fixed
points is not necessarily G-invariant. (The set of attracting fixed points is the right
set to consider, since the set of repelling fixed points is not necessarily a subset of
Λ.) Nevertheless, we have the following:
Proposition 7.4.6. Let G  Isom(X) be nonelementary. Then the set
Λ+ := {g+ : g ∈ G is loxodromic}.
is dense in Λ.
Proof. First note that it suffices to show that Λ+ contains all elements of Λ
which are not global fixed points. Indeed, if this is true, then Λ+ is G-invariant,
and applying Corollary 7.4.3 completes the proof.
Fix ξ ∈ Λ which is not a global fixed point of G, and choose h ∈ G such that

h(ξ) 6= ξ. Fix ε > 0 small enough so that D(B, h(B)) > ε, where B = B(ξ, ε). Let
σ > 0 be large enough so that the Big Shadows Lemma 4.5.7 holds. Since ξ ∈ Λ,

there exists g ∈ G such that

Shad(g(o), σ) ⊆ B.
Let Z = g −1 (Shad(g(o), σ)) = Shadg−1 (o) (o, σ). Then by Lemma 4.5.7, Diam(∂X \
Z) ≤ ε. Thus ∂X \ Z can intersect at most one of the sets B, h(B). So B ⊆ Z or

h(B) ⊆ Z. If B ⊆ Z then

g(B) ⊆ B and B ⊆ Shadg−1 (o) (o, σ),
whereas if h(B) ⊆ Z then
gh(B) ⊆ B and B ⊆ Shad(gh)−1 (o) (o, σ + khk).
So by Lemma 6.3.1, we have j+ ∈ B, where j = g or j = gh is a loxodromic
isometry.

The following improvement over Proposition 7.4.6 has a quite intricate proof:

7.4. MINIMALITY OF THE LIMIT SET

117

Proposition 7.4.7 (Cf. [20, Theorem 5.3.8], [117, p.349]). Let G  Isom(X)

be of general type. Then

{(g+ , g− ) : g ∈ G is loxodromic}
is dense in Λ(G) × Λ(G−1 ). Here G−1 = {g −1 : g ∈ G}.
Proof.
Claim 7.4.8. Let g be a loxodromic isometry and fix ε > 0. There exists
δ = δ(ε, g) such that for all ξ1 , ξ2 ∈ ∂X with D(ξ2 , Fix(g)) ≥ ε,
#{i = 0, . . . , 4 : D(g i (ξ1 ), ξ2 ) ≤ δ} ≤ 1.
Proof. Suppose that D(g i (ξ1 ), ξ2 ) ≤ δ for two distinct values of i. Then

D(g i1 (ξ1 ), g i2 (ξ1 )) ≤ 2δ. For every n, we have

n

D(g n+i1 (ξ1 ), g n+i2 (ξ1 )) .× bkg k δ
and thus by the triangle inequality
D(g i1 (ξ1 ), g n(i2 −i1 )+i1 (ξ1 )) .×,n δ.
By Theorem 6.1.10, if n is sufficiently large then D(g n(i2 −i1 )+i1 (ξ1 ), g+ ) ≤ ε/2,
which implies that

ε/2 ≤ D(ξ2 , Fix(g)) − D(g n(i2 −i1 )+i1 (ξ1 ), g+ ) ≤ D(ξ2 , g n(i2 −i1 )+i1 (ξ1 )) .×,n δ,
which is a lower bound on δ independent of ξ1 , ξ2 . Choosing δ less than this lower
bound yields a contradiction.
⊳
Claim 7.4.9. There exist ε, ρ > 0 such that for all ξ1 , ξ2 , ξ3 , ξ4 ∈ Λ, there exists

j ∈ G such that
(7.4.1)

D(j(ξk ), ξℓ ) ≥ ε ∀k = 1, 2 ∀ℓ = 3, 4 and kjk ≤ ρ.

Proof. Fix g, h ∈ G loxodromic with Fix(g) ∩ Fix(h) = , and let
4

4

ρ = max max kg i hj k.
i=0 j=0

Now fix ξ1 , ξ2 , ξ3 , ξ4 ∈ Λ. By Claim 7.4.8, for each k = 1, 2 and η ∈ Fix(g), we have
#{j = 0, . . . , 4 : D(hj (ξk ), η) ≤ δ1 := δ(D(Fix(g), Fix(h)), h)} ≤ 1.
It follows that there exists j ∈ {0, . . . , 4} such that D(hj (ξk ), η) ≥ δ1 for all k = 1, 2

and η ∈ Fix(g). Applying Claim 7.4.8 again, we see that for each k = 1, 2 and

ℓ = 3, 4, we have

#{i = 0, . . . , 4 : D(g −i (ξℓ ), hj (ξk )) ≤ δ2 := δ(δ1 , g −1 )} ≤ 1.

118

7. LIMIT SETS

It follows that there exists i ∈ {0, . . . , 4} such that D(g −i (ξℓ ), hj (ξk )) ≥ δ2 for all

k = 1, 2 and ℓ = 3, 4. But then

D(g i hj (ξk ), ξℓ ) &× δ2 ,
completing the proof.

⊳

Now fix ξ+ ∈ Λ, ξ− ∈ Λ(G−1 ) distinct, and fix δ > 0 arbitrarily small. By the

definition of Λ, there exist g, h ∈ G such that

D(g(o), ξ+ ), D(h−1 (o), ξ− ) ≤ δ.
Let σ > 0 be large enough so that the Big Shadows Lemma 4.5.7 holds for ε = ε/2,
where ε is as in Claim 7.4.9. Then
Diam(∂X \ Shadg−1 (o) (o, σ)), Diam(∂X \ Shadh(o) (o, σ)) ≤ ε/2.
On the other hand,
Diam(Shad(g(o), σ)), Diam(Shad(h−1 (o), σ)) .× δ.
Since Shad(g(o), σ) is far from h−1 (o) and Shad(h−1 (o), σ) is far from g(o), the
Bounded Distortion Lemma 4.5.6 gives
Diam(h(Shad(g(o), σ))), Diam(g −1 (Shad(h−1 (o), σ))) .× δ.
Choose ξ1 ∈ h(Shad(g(o), σ)), ξ2 ∈ g −1 (Shad(h−1 (o), σ)), ξ3 ∈ ∂X\Shadg−1 (o) (o, σ)

and ξ4 ∈ ∂X \ Shadh(o) (o, σ). By Claim 7.4.9, there exists j ∈ G such that (7.4.1)
holds. Then
Diam(jh(Shad(g(o), σ))), Diam(j −1 g −1 (Shad(h−1 (o), σ))) .× δ,
and by choosing δ sufficiently small, we can make these diameters less than ε/2. It
follows that
jh(Shad(g(o), σ)) ⊆ Shadg−1 (o) (o, σ), and

j −1 g −1 (Shad(h−1 (o), σ)) ⊆ Shadh(o) (o, σ)
or equivalently that

gjh(Shad(g(o), σ)) ⊆ Shad(g(o), σ), and

(gjh)−1 (Shad(h−1 (o), σ)) ⊆ Shad(h−1 (o), σ).

By Lemma 6.3.1, it follows that gjh is a loxodromic isometry satisfying
(gjh)+ ∈ Shad(g(o), σ), (gjh)− ∈ Shad(h−1 (o), σ).
In particular D((gjh)+ , ξ+ ), D((gjh)− , ξ− ) .× δ. Since δ was arbitrary, this completes the proof.



7.5. CONVEX HULLS

119

7.5. Convex hulls
In this section, we assume that X is regularly geodesic (see Section 4.4). Recall
that for points x, y ∈ bord X, the notation [x, y] denotes the geodesic segment, line,
or ray joining x and y.

Definition 7.5.1. Given S ⊆ bord X, let
[
Hull1 (S) :=
[x, y],
x,y∈S

Hulln (S) := Hull1 · · · Hull1 (S)
{z
}
|

Hull∞ (S) :=

[

n times

Hulln (S).

n∈N

The set Hulln (S) will be called the nth convex hull of S. Moreover, Hull∞ (S) will
be called the convex hull of S, and Hull1 (S) will be called the quasiconvex hull of
S.
The terminology “convex hull” comes from the following fact:
Proposition 7.5.2. Hull∞ (S) is the smallest closed set F ⊆ bord X such that

S ⊆ F and
(7.5.1)

[x, y] ⊆ F ∀x, y ∈ F.

A set F satisfying (7.5.1) will be called convex.
Proof. It is clear that S ⊆ Hull∞ (S) ⊆ bord X. To show that Hull∞ (S)
is convex, fix x, y ∈ Hull∞ (S). Then there exist sequences A ∋ xn → x and
S
A ∋ yn → y, where A = n∈N Hulln (S). For each n, [xn , yn ] ⊆ A ⊆ Hull∞ (S). But
since X is regularly geodesic, [xn , yn ] → [x, y] in the Hausdorff metric on bord X.

Since Hull∞ (S) is closed, it follows that [x, y] ⊆ Hull∞ (S).
Conversely, if S ⊆ F ⊆ bord X is a closed convex set, then an induction

argument shows that F ⊇ Hulln (S) for all n. Since F is closed, we have F ⊇
Hull∞ (S).

Another connection between the operations Hull1 and Hull∞ is given by the
following proposition:
Proposition 7.5.3. Suppose that X is a algebraic hyperbolic space. Then there
exists τ > 0 such that for every set S ⊆ bord X we have
X ∩ Hull1 (S) ⊆ X ∩ Hull∞ (S) ⊆ Nτ (X ∩ Hull1 (S)).
(Recall that Nτ (S) denotes the τ -thickening of a set with respect to the hyperbolic
metric d.)

120

7. LIMIT SETS

Proof. The proof will proceed using the ball model X = B = Bα
F . We will
need the following lemma:
Lemma 7.5.4. There exists a closed convex set F $ bord B whose interior
intersects ∂B.
Proof. If α < ∞, this is a consequence of [11, Theorem 3.3].
We will use the finite-dimensional case to prove the infinite-dimensional case.
Suppose that α is infinite. Let Y = B3F ⊆ X. Then by the α < ∞ case of Lemma
7.5.4, there exists a closed convex set F2 $ bord Y whose interior intersects ∂Y ,
say ξ ∈ Int(F2 ) ∩ ∂Y . Choose ε > 0 such that BY (ξ, ε) ⊆ F2 . Then
F1 := Hull∞ (BY (ξ, ε)) ⊆ F2 $ bord Y
by Proposition 7.5.2. On the other hand, F1 is invariant under the action of the
group
G1 := {g ∈ Isom(Y ) : g(0) = 0, g(ξ) = ξ}.
Let
G = {g ∈ Isom(X) : g(0) = 0, g(ξ) = ξ},
and note that G(bord Y ) = bord X. Let F = G(F1 ), and note that F ∩bord Y = F1 .

We claim that F is convex. Indeed, suppose that x, y ∈ F ; then there exists g ∈ G
such that g(x), g(y) ∈ bord Y . (Note that in this step, we need all three dimensions

of Y .) Then g(x), g(y) ∈ F ∩ bord Y = F1 , so by the convexity of F1 we have
g([x, y]) = [g(x), g(y)] ⊆ F1 ⊆ F . Since F is G-invariant, we have [x, y] ⊆ F .
In addition to being convex, F is also closed and contains the set G(BY (ξ, ε)) =
BX (ξ, ε). Thus, ξ ∈ Int(F ). Finally, since F ∩ bord Y = F1 $ bord Y , it follows

that F $ bord X.

⊳

Let F be as in Lemma 7.5.4. Since F $ bord B is a closed set, it follows that
B \ F 6= . By the transitivity of Isom(B) (Observation 2.3.2), we may without loss
of generality assume that 0 ∈ B \ F . By the transitivity of Stab(Isom(B); 0) on ∂B,
we may without loss of generality assume that e1 ∈ Int(F ). Fix ε > 0 such that
B(e1 , ε) ⊆ F .
We now proceed with the proof of Proposition 7.5.3. It is clear from the definitions that B ∩ Hull1 (S) ⊆ B ∩ Hull∞ (S). To prove the second inclusion, fix

z ∈ B \ Nτ (Hull1 (S)) and we will show that z ∈
/ Hull∞ (S). By the transitivity
of Isom(B), we may without loss of generality assume that z = 0. Now for every
x, y ∈ S, we have z = 0 ∈
/ Nτ ([x, y]). By (i) of Proposition 4.3.1, we have
hx|yi0 &+ τ
and thus by (3.5.1),
ky − xk .× e−τ .

7.5. CONVEX HULLS

121

By choosing τ sufficiently large, this implies that
ky − xk ≤ ε/2 ∀x, y ∈ S.
Moreover, since d(0, x) = 2hx|xi0 &+ τ , by choosing τ sufficiently large we may
guarantee that
kxk ≥ 1 − ε/2 ∀x ∈ S.
Since the claim is trivial if S = , assume that S 6=  and choose x ∈ S. Without

loss of generality, assume that x = λe1 for some λ ≥ 2/3. Then S ⊆ BE (x, ε/2) ⊆
B(e1 , ε) ⊆ F . But then F is a closed convex set containing S, so by Proposition
7.5.2 Hull∞ (S) ⊆ F . Since z = 0 ∈
/ F , it follows that z ∈
/ Hull∞ (S).



Corollary 7.5.5. Suppose that X is an algebraic hyperbolic space. Then for
every closed set S ⊆ bord X, we have
Hull∞ (S) ∩ ∂X = S ∩ ∂X.
Proof. The inclusion ⊇ is immediate. Suppose that ξ ∈ Hull∞ (S) ∩ ∂X, and
find a sequence X ∩ Hull∞ (S) ∋ xn → ξ. By Proposition 7.5.3, for each n there
(1)

(2)

exist yn , yn
(1)

(2)

(1)

(2)

∈ S such that xn ∈ Nτ ([yn , yn ]); by Proposition 4.3.1 we have

hyn |yn ixn ≍+ 0. Applying Lemma 7.4.2 gives ξ ∈ S.



Remark 7.5.6. Corollary 7.5.5 was proven for the case where X is a pinched
(finite-dimensional) Hadamard manifold and S ⊆ ∂X by M. T. Anderson [11,

Theorem 3.3]. It was conjectured to hold whenever X is “strictly convex” by
Gromov [83, p.11], who observed that it holds in the Standard Case. However,

this conjecture was proven to be false independently by A. Ancona [9, Corollary
C] and A. Borbély [32, Theorem 1], who each constructed a three-dimensional
CAT(-1) manifold X and a point ξ ∈ ∂X such that for every neighborhood U of ξ,
Hull∞ (U ) = bord X.
Thus, although the ∞-convex hull has more geometric and intuitive appeal
based on Proposition 7.5.2, without more hypotheses there is no way to restrain its
geometry. The 1-convex hull is thus more useful for our applications. Proposition
7.5.3 indicates that in the case of an algebraic hyperbolic space, we are not losing
too much by the change.
Definition 7.5.7. The convex core of a semigroup G  Isom(X) is the set
CΛ := X ∩ Hull∞ (Λ),
and the quasiconvex core is the set
Co := X ∩ Hull1 (G(o)).

122

7. LIMIT SETS

Observation 7.5.8. The convex core and quasiconvex core are both closed
G-invariant sets. The quasiconvex core depends on the distinguished point o. However:
Proposition 7.5.9. Fix x, y ∈ X. Then
Cx ⊆ NR (Cy )
for some R > 0.
Proof. Fix z ∈ Cx . Then z ∈ [g(x), h(x)] for some g, h ∈ G. It follows that
hg(y)|h(y)iz ≍+ hg(x)|h(x)iz = 0.
So by Proposition 4.3.1, d(z, [g(y), h(y)]) ≍+ 0. But [g(y), h(y)] ⊆ Cy , so d(z, Cy ) ≍+
0. Letting R be the implied constant completes the proof.

Remark 7.5.10. In many cases, we can get information about the action of G
on X by looking just at its restriction to CΛ or to Co . We therefore also remark
that if X is a CAT(-1) space, then CΛ is also a CAT(-1) space.
In the sequel the following notation will be useful:
Notation 7.5.11. For a set S ⊆ bord X let
S ′ = S ∩ ∂X.

(7.5.2)

Observation 7.5.12. (Co )′ = Λ.
Proof. Since Λ = (G(o))′ and G(o) ⊆ Co , we have (Co )′ ⊇ Λ. Suppose that
(1) (2)
ξ ∈ (Co )′ , and let Co ∋ xn → ξ. By definition, for each n there exist yn , yn ∈ G(o)
(1)

(2)

such that xn ∈ [yn , yn ]. Lemma 7.4.2 completes the proof.



7.6. Semigroups which act irreducibly on algebraic hyperbolic spaces
Definition 7.6.1. Suppose that X is an algebraic hyperbolic space, and fix
G  Isom(X). We shall say that G acts reducibly on X if there exists a nontrivial
totally geodesic G-invariant subset S $ bord X. Otherwise, we shall say that G
acts irreducibly on X.

Remark 7.6.2. A parabolic or focal subsemigroup of Isom(X) may act either
reducibly or irreducibly on X.
Proposition 7.6.3. Let G  Isom(X) be nonelementary. Then the following

are equivalent:

(A) G acts reducibly on X.

7.7. SEMIGROUPS OF COMPACT TYPE

123

(B) There exists a nontrivial totally geodesic subset S $ bord X such that
Λ ⊆ S.
(C) There exists a nontrivial totally geodesic subset S $ bord X such that
CΛ ⊆ S.
(D) There exists a nontrivial totally geodesic subset S $ bord X such that
Co ⊆ S for some o ∈ X.
Proof of (A) ⇒ (B). Let S $ bord X be a nontrivial totally geodesic Ginvariant subset. Fix o ∈ S ∩ X. Then Λ ⊆ G(o) ⊆ S.

Proof of (B) ⇒ (C). If S is any totally geodesic set which contains Λ, then
S is a closed convex set containing Λ, so by Proposition 7.5.2, CΛ ⊆ S.

Proof of (C) ⇒ (D). Since G is nonelementary, CΛ 6= . Fix o ∈ CΛ ; then

Co ⊆ CΛ .



Proof of (D) ⇒ (A). Let S be the smallest totally geodesic subset of X

which contains Co , i.e.

S :=

\
{W : W ⊇ Co totally geodesic}.

Then our hypothesis implies that S $ bord X. Since o ∈ S, S is nontrivial. It is
obvious from the definition that S is G-invariant. This completes the proof.

Remark 7.6.4. If G  Isom(X) is nonelementary, then Proposition 7.6.3 gives

us a way to find a nontrivial totally geodesic set on which G acts reducibly; namely,
the smallest totally geodesic set containing Λ, or equivalently CG , will have this

property (cf. Lemma 2.4.5). On the other hand, there exists a parabolic group
G ≤ Isom(H∞ ) such that G does not act irreducibly on any nontrivial totally

geodesic subset S ⊆ bord H∞ (Remark 11.2.19).

7.7. Semigroups of compact type
Definition 7.7.1. We say that a semigroup G  Isom(X) is of compact type

if its limit set Λ is compact.

Proposition 7.7.2. For G  Isom(X), the following are equivalent:
(A) G is of compact type.
(B) Every sequence (xn )∞
1 in G(o) with kxn k → ∞ has a convergent subsequence.

Furthermore, if X is regularly geodesic, then (A)-(B) are equivalent to:
(C) The set Co is a proper metric space.
and if X is an algebraic hyperbolic space, then they are equivalent to:

124

7. LIMIT SETS

(D) The set CΛ is a proper metric space.
Proof of (A) ⇒ (B). Fix a sequence (gn )∞
1 in G with kgn k → ∞. The exis-

tence of such a sequence implies that G is not elliptic. If G is parabolic or inward
focal, then the proof of Proposition 7.3.1(ii) shows that gn (o) → ξ, where Λ = {ξ}.

So we may assume that G is lineal, outward focal, or of general type, in which case
Proposition 7.3.1 gives #(Λ) ≥ 2.

∞
Fix distinct ξ1 , ξ2 ∈ Λ, and let (nk )∞
1 be a sequence such that (gnk (ξi ))1
converges for i = 1, 2, and such that

hgn−1
(o)|ξ1 io ≤ hgn−1
(o)|ξ2 io
k
k
for all k. (If this is not possible, switch ξ1 and ξ2 .) We have

(o)|ξ1 io
(o)|ξ1 io , hgn−1
(o)|ξ2 io = hgn−1
0 ≍+,ξ1 ,ξ2 hξ1 |ξ2 io &+ min hgn−1
k
k
k

and thus

→ ∞.
hgnk (o)|gnk (ξ1 )io ≍+,ξ1 ,ξ2 kgnk k −
n

On the other hand, there exists η ∈ Λ such that gnk (ξ1 ) −
→ η, and thus
k

hgnk (ξ1 )|ηio −
→ ∞.
n

Applying Gromov’s inequality yields
hgnk (o)|ηio −
→∞
n

and thus gnk (o) −
→ η. This completes the proof.



k

Proof of (B) ⇒ (A). Fix a sequence (ξn )∞
1 in Λ. For each n ∈ N, choose
gn ∈ G with
hgn (o)|ξn io ≥ n.

In particular kgn k ≥ n −
→ ∞. Thus by our hypothesis, there exists a convergent
n

subsequence gnk (o) −
→ η ∈ Λ. Now
k

D(ξnk , η) ≤ D(gnk (o), ξnk ) + D(gnk (o), η) .× b−nk + D(gnk (o), η) −
→ 0,
k

i.e. ξnk −
→ η.



k

Proof of (A) ⇒ (C). Let (xn )∞
1 be a bounded sequence in Co . For each
(1)

(1)

(2)

(2)

n ∈ N, there exist yn , yn ∈ G(o) such that xn ∈ N1/n ([yn , yn ]). Choose a
(1)
(2)
→ α and ynk −
→ β. Since X is regularly geodesic we
sequence (nk )∞
1 on which ynk −

have

k

k

[yn(1)
, yn(2)
]−
→ [y (1) , y (2) ].
k
k
k

For each k, choose zk ∈

(1) (2)
[ynk , ynk ]

with d(xnk , zk ) ≤ 1/nk . Since the sequence (zk )∞
1

is bounded, it must have a subsequence which converges to a point in [y (1) , y (2) ];

7.7. SEMIGROUPS OF COMPACT TYPE

125

it follows that the corresponding subsequence of (xnk )∞
1 is also convergent. Thus
every bounded sequence in Co has a convergent subsequence, so Co is proper.
Proof of (C) ⇒ (B). Obvious since G(o) ⊆ Co .




Proof of (A) ⇒ (D). Note first of all that we cannot get (A) ⇒ (D) immediately from Proposition 7.5.3, since the τ -thickening of a compact set is no longer
compact.
By [35, Proposition 1.5], there exists a metric ρ on bord X compatible with the
topology such that the map F 7→ Hull1 (F ) is a semicontraction with respect to the
Hausdorff metric of (bord X, ρ). (Finite-dimensionality is not used in any crucial
way in the proof of [35, Proposition 1.5],2 and in any case for algebraic hyperbolic
spaces it can be proven by looking at finite-dimensional subsets, as we did in the
proof of Proposition 7.5.3.) We remark that if F = R, then such a metric ρ can
be prescribed explicitly: if X = B is the ball model, then the Euclidean metric on
bord B ⊆ H has this property, due to the fact that geodesics in the ball model are
line segments in H (cf. (2.2.3)).3
Now let us demonstrate (D). It suffices to show that bord CΛ = Hull∞ (Λ) is

compact. Since Hull∞ (Λ) is by definition closed, it suffices to show that Hull∞ (Λ)
is totally bounded with respect to the ρ metric. Indeed, fix ε > 0. Since Λ is
compact, there is a finite set Fε ⊆ Λ such that
Λ ⊆ Nε/2 (Fε ).
(In this proof, all neighborhoods are taken with respect to the ρ metric.) Let Xε ⊆
X be a finite-dimensional totally geodesic set containing Fε . Then Λ ⊆ Nε/2 (Xε ).
On the other hand, since Xε is compact, there exists a finite set Fε′ ⊆ Xε such that
Xε ⊆ Nε/2 (Fε′ ).
Now, our hypothesis on ρ implies that

Hull1 (Nε/2 (Xε )) ⊆ Nε/2 (Hull1 (Xε )) = Nε/2 (Xε ),
and thus that Nε/2 (Xε ) is convex. But Λ ⊆ Nε/2 (Xε ), so Hull∞ (Λ) ⊆ Nε/2 (Xε ).
Thus
Hull∞ (Λ) ⊆ Nε/2 (Xε ) ⊆ Nε (Fε′ ).
Since ε was arbitrary, this shows that Hull∞ (Λ) is totally bounded, completing the
proof.

Proof of (D) ⇒ (B). Since property (B) is clearly basepoint-independent,

we may without loss of generality suppose o ∈ CΛ . Then (D) ⇒ (C) ⇒ (B).



2One should keep in mind that the Cartan–Hadamard theorem [119, IX, Theorem 3.8] can be
used as a substitute for the Hopf–Rinow theorem in most circumstances.
3Recall that our “ball model” B is the Klein model rather than the Poincaré model.

126

7. LIMIT SETS

As an example of an application we prove the following corollary.
Corollary 7.7.3. Suppose that X is regularly geodesic. Then any moderately
discrete subgroup of Isom(X) of compact type is strongly discrete.
Proof. If G is a moderately discrete group, then G ↿ Co is moderately discrete

by Observation 5.2.14, and therefore strongly discrete by Propositions 5.2.5 and
7.7.2. Thus by Observation 5.2.14, G is strongly discrete.

A well-known characterization of the complement of the limit set in the Standard Case is that it is the set of points where the action of G is discrete. We extend
this characterization to hyperbolic metric spaces for groups of compact type:
Proposition 7.7.4. Let G ≤ Isom(X) be a strongly discrete group of compact
type. Then the action of G on bord X \ Λ is strongly discrete in the following sense:
For any set S ⊆ bord X \ Λ satisfying

(7.7.1)

D(S, Λ) > 0,

we have
#{g ∈ G : g(S) ∩ S 6= } < ∞.
Proof. By contradiction, suppose that there exists a sequence of distinct
(gn )∞
1 such that gn (S) ∩ S 6=  for all n ∈ N. Since G is strongly discrete, we

have kgn k → ∞, and since G is of compact type there exist an increasing sequence

−1
(nk )∞
1 and ξ+ , ξ− ∈ Λ such that gnk (o) → ξ+ and gnk (o) → ξ− . In the remainder
of the proof we restrict to this subsequence, so that gn (o) → ξ+ and gn−1 (o) → ξ− .

For each n, fix xn ∈ gn−1 (gn (S) ∩ S), so that xn , gn (xn ) ∈ S. Then
D(xn , ξ− ), D(gn (xn ), ξ+ ) ≥ D(S, Λ) ≍× 1,

and so
hxn |ξ− io , hgn (xn )|ξ+ io ≍+ 0.

On the other hand, hgn−1 (o)|ξ− io , hgn (o)|ξ+ io → ∞. Applying Gromov’s inequality
gives
hxn |gn−1 (o)io , hgn (xn )|gn (o)io ≍+ 0
for all n sufficiently large. But then
kgn k = hgn (xn )|oign (o) + hgn (xn )|gn (o)io ≍+ 0,
a contradiction.



Part 2

The Bishop–Jones theorem

This part will be divided as follows: In Chapter 8, we motivate and define
the modified Poincaré exponent of a semigroup, which is used in the statement of
Theorem 1.2.3. In Chapter 9 we prove Theorem 1.2.3 and deduce Theorem 1.2.1
from Theorem 1.2.3.

CHAPTER 8

The modified Poincaré exponent
In this chapter we define the modified Poincaré exponent of a semigroup. We
first recall the classical notion of the Poincaré exponent, introduced in the Standard
Case by A. F. Beardon in [18]. Although it is usually defined only for groups, the
generalization to semigroups is trivial.

8.1. The Poincaré exponent of a semigroup
Definition 8.1.1. Fix G  Isom(X). For each s ≥ 0, the series
X
Σs (G) :=
b−skgk
g∈G

is called the Poincaré series of the semigroup G in dimension s (or “evaluated at
s”) relative to b. The number
δG = δ(G) := inf{s ≥ 0 : Σs (G) < ∞}
is called the Poincaré exponent of the semigroup G relative to b. Here, we let
inf  = ∞.
Remark 8.1.2. The Poincaré series is usually defined with a summand of e−skgk
rather than b−skgk . The change of exponents here is important because it relates
the Poincaré exponent to the metric D = Db,o defined in Proposition 3.6.8. In
the Standard Case, and more generally for CAT(-1) spaces, we have made the
convention that b = e (see §4.1), so in this case our series reduces to the classical
one.

Remark 8.1.3. Given G  Isom(X), we may define the orbital counting func-

tion of G to be the function

NX,G (ρ) = #{g ∈ G : kgk ≤ ρ}.
129

130

8. THE MODIFIED POINCARÉ EXPONENT

The Poincaré series may be written as an integral over the orbital counting function
as follows:
Σs (G) = log(bs )
(8.1.1)

= log(bs )

XZ

g∈G
∞

Z

0

s

= log(b )

Z

∞

b−sρ dρ

kgk

b−sρ

X

g∈G
∞

0

[kgk ≤ ρ] dρ

b−sρ NX,G (ρ) dρ.

The Poincaré exponent is written in terms of the orbital counting function as
(8.1.2)

δG = lim sup
ρ→∞

1
logb NX,G (ρ)
ρ

Definition 8.1.4. A semigroup G  Isom(X) with δG < ∞ is said to be of
convergence type if ΣδG (G) < ∞. Otherwise, it is said to be of divergence type. In
the case where δG = ∞, we say that the semigroup is neither of convergence type
nor of divergence type.

The most basic question about the Poincaré exponent is whether it is finite.
For groups, the finiteness of the Poincaré exponent is related to strong discreteness:
Observation 8.1.5. Fix G ≤ Isom(X). If G is not strongly discrete, then

δG = ∞.

Proof. Fix ρ > 0 such that #{g ∈ G : kgk ≤ ρ} = ∞. Then for all s ≥ 0 we
have
X
X
Σs (G) ≥
b−skgk ≥
b−sρ = ∞.
g∈G
kgk≤ρ

g∈G
kgk≤ρ

Since s was arbitrary, we have δG = ∞.



Remark 8.1.6. Although the converse to Observation 8.1.5 holds in the Standard Case, it fails for infinite-dimensional algebraic hyperbolic spaces; see Example
13.2.2.
Notation 8.1.7. The Poincaré exponent and type can be conveniently combined into a single mathematical object, the Poincaré set



[0, δG ] G is of divergence type


∆G := {s ≥ 0 : Σs (G) = ∞} = [0, δG ) G is of convergence type .



[0, ∞) δ = ∞
G

8.2. The modified Poincaré exponent of a semigroup

From a certain perspective, Observation 8.1.5 indicates a flaw in the Poincaré
exponent: If G ≤ Isom(X) is not strongly discrete, then the Poincaré exponent is

8.2. THE MODIFIED POINCARÉ EXPONENT OF A SEMIGROUP

131

always infinity even though there may be more geometric information to capture.
In this section we introduce a modification of the Poincaré exponent which agrees
with the Poincaré exponent in the case where G is strongly discrete, but can be
finite even if G is not strongly discrete.
We begin by defining the modified Poincaré exponent of a locally compact
group G ≤ Isom(X). Let µ be a Haar measure on G, and for each s consider the
Poincaré integral
Z
(8.2.1)
Is (G) := b−skgk dµ(g).
Definition 8.2.1. The modified Poincaré exponent of a locally compact group
G ≤ Isom(X) is the number
e
δeG = δ(G)
:= inf{s ≥ 0 : Is (G) < ∞},

where Is (G) is defined by (8.2.1).

Example 8.2.2. Let X = Hd for some 2 ≤ d < ∞, and let G ≤ Isom(X) be

a positive-dimensional Lie subgroup. Then G is locally compact, but not strongly
discrete. Although the Poincaré series diverges for every s, the exponent of convergence of the Poincaré integral (or “modified Poincaré exponent”) is equal to
the Hausdorff dimension of the limit set of G (Theorem 1.2.3 below), and so in
particular the Poincaré integral converges whenever s > d − 1.
We now proceed to generalize Definition 8.2.1 to the case where G ≤ Isom(X)
is not necessarily locally compact. Fix ρ > 0, and consider a maximal ρ-separated1
subset Sρ ⊆ G(o). Then we have
[
[
B(x, ρ/2) ⊆ G(o) ⊆
B(x, ρ),
x∈Sρ

x∈Sρ

and the former union is disjoint. Now suppose that G is in fact locally compact,
and let ν denote the image of Haar measure on G under the map g 7→ g(o). Then

if f is a positive function on X whose logarithm is uniformly continuous, we have
Z
X
XZ
XZ
X
f (x) ≍×,ρ,f
f dν ≤ f dν ≤
f dν ≍×,ρ,f
f (x).

x∈Sρ

x∈Sρ

B(x,ρ/2)

x∈Sρ

B(x,ρ)

x∈Sρ

Thus in some sense, the counting measure on Sρ is a good approximation to the
measure ν. In particular, taking f (x) = b−kxk gives
X
Is (G) ≍×,ρ
b−kxk .
x∈Sρ

1Here, as usual, a ρ-separated subset of a metric space X is a set S ⊆ X such that d(x, y) ≥ ρ
for any distinct x, y ∈ S. The existence of a maximal ρ-separated subset of any metric space is
guaranteed by Zorn’s lemma.

132

8. THE MODIFIED POINCARÉ EXPONENT

Thus the integral Is (G) converges if and only if the series

P

x∈Sρ

b−kxk converges.

But the latter series is well-defined even if G is not locally compact. This discussion
shows that the definition of the “modified Poincaré exponent” given in Definition
8.2.1 agrees with the following definition:
Definition 8.2.3. Fix G  Isom(X).
• For each set S ⊆ X and s ≥ 0, let
X
Σs (S) =
b−skxk
x∈S

∆(S) = {s ≥ 0 : Σs (S) = ∞}
δ(S) = sup ∆(S).

• Let
eG =
∆

(8.2.2)

\\

∆(Sρ ),

ρ>0 Sρ

where the second intersection is taken over all maximal ρ-separated sets
Sρ .
e G is called the modified Poincaré exponent of
• The number δeG = sup ∆
e
e
G. If δG ∈ ∆G , we say that G is of generalized divergence type,2 while if
δeG ∈ [0, ∞) \ ∆G , we say that G is of generalized convergence type. Note
that if δeG = ∞, then G is neither of generalized convergence type nor of
generalized divergence type.

The basic properties of the modified Poincaré exponent are summarized as
follows:
Proposition 8.2.4. Fix G  Isom(X).

e G ⊆ ∆G . (In particular δeG ≤ δG .)
(i) ∆
(ii) If G satisfies

(8.2.3)

sup #{g ∈ G : d(g(o), x) ≤ ρ} < ∞ ∀ρ > 0,

x∈X

e G = ∆G . (In particular δeG = δG .)
then ∆
(iii) If δeG < ∞, then there exist ρ > 0 and a maximal ρ-separated set Sρ ⊆ G(o)
such that #(Sρ ∩ B) < ∞ for every bounded set B.

(iv) For all ρ > 0 sufficiently large and for every maximal ρ-separated set
e
e G . (In particular δ(Sρ ) = δ(G).)
Sρ ⊆ G(o), we have ∆(Sρ ) = ∆

2We use the adjective “generalized” rather than “modified” because all groups of conver-

gence/divergence type are also of generalized convergence/divergence type; see Corollary 8.2.8
below.

8.2. THE MODIFIED POINCARÉ EXPONENT OF A SEMIGROUP

133

Remark 8.2.5. If G is a group, then it is clear that (8.2.3) is equivalent to
the assertion that G is strongly discrete. If G is not a group, then by analogy we
will say that G is strongly discrete if (8.2.3) holds. (Recall that in Chapter 5, the
various notions of discreteness are defined only for groups.)
Proof of Proposition 8.2.4.
(i) Indeed, for every s ≥ 0, ρ > 0, and maximal ρ-separated set Sρ we have
e
Σs (Sρ ) ≤ Σs (G) and thus ∆(G)
⊆ ∆(Sρ ) ⊆ ∆(G).

(ii) Fix ρ > 0, and let Sρ ⊆ G(o) be a maximal ρ-separated set. For every
x ∈ G(o) there exists yx ∈ Sρ with d(x, yx ) ≤ ρ. Then for each y ∈ Sρ ,
we have

#{x ∈ G(o) : yx = y} ≤ Mρ ,
where Mρ is the value of the supremum (8.2.3). Therefore for each s ≥ 0

we have
X
X
X
b−skyx k ≤ Mρ
b−skxk ≍×
b−skyk = Mρ Σs (Sρ ).
Σs (G) =
x∈G(o)

x∈G(o)

y∈Sρ

In particular, Σs (G) < ∞ if and only if Σs (Sρ ) < ∞, i.e. ∆(G) = ∆(Sρ ).
e
Intersecting over ρ > 0 and Sρ ⊆ G(o) yields ∆(G) = ∆(G).

(iii) Take ρ and Sρ such that δ(Sρ ) < ∞.
Before proving (iv), we need a lemma:

Lemma 8.2.6. Fix ρ1 , ρ2 > 0 with ρ2 ≥ 2ρ1 . Let S1 ⊆ G(o) be a ρ1 -net,3 and
let S2 ⊆ G(o) be a ρ2 -separated set. Then
∆(S2 ) ⊆ ∆(S1 ).

(8.2.4)

Proof. Since S1 is a ρ1 -net, for every y ∈ S2 , there exists xy ∈ S1 with

d(y, xy ) < ρ1 . If xy = xz for some y, z ∈ S2 , then d(y, z) < 2ρ1 ≤ ρ2 and since S2
is ρ2 -separated we have y = z. Thus the map y 7→ xy is injective. It follows that
for every s ≥ 0, we have
X
X
X
Σs (S2 ) =
b−skyk ≍×
b−skxy k ≤
b−skxk = Σs (S1 ),
y∈S2

demonstrating (8.2.4).

y∈S2

x∈S1

⊳

(iv) The statement is trivial if δeG = ∞. So suppose that δeG < ∞, and let ρ, Sρ
be as in (iii). Fix ρ′ ≥ 2ρ and a maximal ρ′ -separated set Sρ′ ⊆ G(o), and
e G . The inclusion ⊇ follows by definition.
we will show that ∆(Sρ′ ) = ∆

3Here, as usual, a ρ-net in a metric space X is a subset S ⊆ X such that X = N (S). Note that
ρ
every maximal ρ-separated set is a ρ-net (but not conversely).

134

8. THE MODIFIED POINCARÉ EXPONENT

To prove the reverse direction, fix ρ′′ > 0 and a maximal ρ′′ -separated set
Sρ′′ , and we will show that ∆(Sρ′ ) ⊆ ∆(Sρ′′ ).
Let F = Sρ ∩ B(o, ρ′′ + ρ); then #(F ) < ∞. We then set
[
Sρ′ :=
gx (F )
x∈Sρ′′

where for each x ∈ Sρ′′ , x = gx (o). Then for all s ≥ 0,
X
X X
b−skxk ≍×
Σs (Sρ′′ ) =
b−skxk
x∈Sρ′′

x∈Sρ′′ y∈F

≍×

X X

b−skgx (y)k

x∈Sρ′′ y∈F

= Σs (Sρ′ )

and therefore ∆(Sρ′′ ) = ∆(Sρ′ ). But Sρ′ is a ρ-net, so by Lemma 8.2.6, we
have ∆(Sρ′ ) ⊆ ∆(Sρ′ ). This completes the proof.

Combining with Observation 8.1.5 yields the following:
e then
Corollary 8.2.7. Suppose that G is a group. If ∆ 6= ∆
δe < δ = ∞.

Corollary 8.2.8. If a group G is of convergence or divergence type, then it
is also of generalized convergence or divergence type, respectively.
e G = ∆G , and Poincaré
We will call a group G ≤ Isom(X) Poincaré regular if ∆

irregular otherwise. A list of sufficient conditions for Poincaré regularity is given in
Proposition 9.3.1 below. Conversely, several examples of Poincaré irregular groups
may be found in Section 13.4.

CHAPTER 9

Generalization of the Bishop–Jones theorem
In this chapter we prove Theorem 1.2.3, the first part of which states that if
G  Isom(X) is a nonelementary semigroup, then
(1.2.2)

dimH (Λr ) = dimH (Λur ) = dimH (Λur ∩ Λr,σ ) = δe

for some σ > 0. Our strategy is to prove that dimH (Λur ∩ Λr,σ ) ≤ dimH (Λur ) ≤
dimH (Λr ) ≤ δe ≤ dimH (Λur ∩ Λr,σ ) for some σ > 0. The first two inequalities are
obvious. The third we prove now, and the proof of the fourth inequality will occupy
§§9.1-9.2.

Lemma 9.0.9. For G  Isom(X), we have
e
dimH (Λr ) ≤ δ.

e we have
Proof. It suffices to show that for each σ > 0 and for each s > δ,
e
dimH (Λr,σ ) ≤ s. Fix σ > 0 and s > δ. Then there exists ρ > 0 and a maximal
ρ-separated set Sρ ⊆ G(o) such that s > δ(Sρ ), which implies that Σs (Sρ ) < ∞.
For each x ∈ Sρ let Px = Shad(x, σ + ρ).
Claim 9.0.10.
ξ ∈ Λr,σ ⇒ ξ ∈ Px for infinitely many x ∈ Sρ .
Proof. Fix ξ ∈ Λr,σ . Then there exists a sequence gn (o) → ξ such that for
all n ∈ N we have ξ ∈ Shad(gn (o), σ). For each n, let xn ∈ Sρ be such that

d(gn (o), xn ) ≤ ρ; such an xn exists since Sρ is maximal ρ-separated. Then by (d)
of Proposition 3.3.3 we have ξ ∈ Pxn = Shad(xn , σ + ρ).

To complete the proof of Claim 9.0.10, we need to show that the collection
(xn )∞
1 is infinite. Indeed, if xn ∈ F for some finite F and for all n ∈ N, then we

would have d(gn (o), F ) ≤ ρ for all n ∈ N. This would imply that the sequence
⊳
(gn (o))∞
1 is bounded, contradicting that gn (o) → ξ.
We next observe that by the Diameter of Shadows Lemma 4.5.8 we have
X
X
Diams (Px ) .×,σ,ρ
b−skxk = Σs (Sρ ) < ∞.
x∈Sρ

x∈Sρ
135

136

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

Thus by the Hausdorff–Cantelli lemma [24, Lemma 3.10], we have Hs (Λr,σ ) = 0,
and thus dimH (Λr,σ ) ≤ s.



9.1. Partition structures
In this section we introduce the notion of a partition structure, an important
technical tool for proving Theorem 1.2.3. We state some theorems about these
structures, which will be proven in subsequent sections, and then use them to
prove Theorem 1.2.3. 1
Throughout this section, (Z, D) denotes a metric space. We will constantly
have in mind the special case Z = ∂X, D = Db,o .
Notation 9.1.1. Let
N∗ =

∞
[

Nn .

n=0

If ω ∈ N∗ ∪ NN , then we denote by |ω| the unique element of N ∪ {∞} such that

ω ∈ N|ω| and call |ω| the length of ω. For each r ∈ N, we denote the initial segment
of ω of length r by
ω1r := (ωn )r1 ∈ Nr .
For two words ω, τ ∈ NN , let ω ∧ τ denote their longest common initial segment,
and let
d2 (ω, τ ) = 2−|ω∧τ |.
Then (NN , d2 ) is a metric space.
Definition 9.1.2. A tree on N is a set T ⊆ N∗ which is closed under initial
segments. (Not to be confused with the various notions of “trees” introduced in
Section 3.1.)
Notation 9.1.3. If T is a tree on N, then we denote its set of infinite branches
by
T (∞) := {ω ∈ NN : ω1n ∈ T ∀n ∈ N}.
On the other hand, for n ∈ N we let
T (n) := T ∩ Nn .
For each ω ∈ T , we denote the set of its children by
T (ω) := {a ∈ N : ωa ∈ T }.
1 Much of the material for this section has been taken (with modifications) from [73, §5]. In [73]

we also included as standing assumptions that G was strongly discrete and of general type (see
Definitions 5.2.1 and 6.2.13). Thus some propositions which appear to have the exact same statement are in fact stronger in this monograph than in [73]. Specifically, this applies to Proposition
9.1.9 and Lemmas 9.2.1 and 9.2.5.

9.1. PARTITION STRUCTURES

137

Definition 9.1.4. A partition structure on Z consists of a tree T ⊆ N∗ together

with a collection of closed subsets (Pω )ω∈T of Z, each having positive diameter and
enjoying the following properties:
(I) If ω ∈ T is an initial segment of τ ∈ T then Pτ ⊆ Pω . If neither ω nor τ

is an initial segment of the other then Pω ∩ Pτ = .
(II) For each ω ∈ T let
Dω = Diam(Pω ).

There exist κ > 0 and 0 < λ < 1 such that for all ω ∈ T and for all
a ∈ T (ω), we have
D(Pωa , Z \ Pω ) ≥ κDω

(9.1.1)
and
(9.1.2)

κDω ≤ Dωa ≤ λDω .

Fix s > 0. The partition structure (Pω )ω∈T is called s-thick if for all ω ∈ T ,
X
s
Dωa
≥ Dωs .
(9.1.3)
a∈T (ω)

Definition 9.1.5. Given a partition structure (Pω )ω∈T , a substructure of
(Pω )ω∈T is a partition structure of the form (Pω )ω∈Te , where Te ⊆ T is a subtree.
Observation 9.1.6. Let (Pω )ω∈T be a partition structure on a complete metric
space (Z, D). For each ω ∈ T (∞), the set
\
Pω1n
n∈N

is a singleton. If we define π(ω) to be the unique member of this set, then the map
π : T (∞) → Z is continuous. (In fact, it was shown in [73, Lemma 5.11] that π is
quasisymmetric.)
Definition 9.1.7. The set π(T (∞)) is called the limit set of the partition
structure.
We remark that a large class of examples of partition structures comes from the
theory of conformal iterated function systems [128] (or in fact even graph directed
Markov systems [129]) satisfying the strong separation condition (also known as the
disconnected open set condition [150]; see also [71], where the limit sets of iterated
function systems satisfying the strong separation condition are called dust-like).
Indeed, the notion of a partition structure was intended primarily to generalize
these examples. The difference is that in a partition structure, the sets (Pω )ω do
not necessarily have to be defined by dynamical means. We also note that if Z = Rd
for some d ∈ N, and if (Pω )ω∈T is a partition structure on Z, then the tree T has

138

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

bounded degree, meaning that there exists N < ∞ such that #(T (ω)) ≤ N for

every ω ∈ T .
We will now state two propositions about partition structures and then use
them to prove Theorem 1.2.3. Theorem 9.1.8 will be proven below, and Proposition
9.1.9 will be proven in the following section.
Theorem 9.1.8 ([73, Theorem 5.12]). Fix s > 0. Then any s-thick partition
structure (Pω )ω∈T on a complete metric space (Z, D) has a substructure (Pω )ω∈Te
whose limit set is Ahlfors s-regular. Furthermore the tree Te can be chosen so that
for each ω ∈ Te, we have that Te(ω) is an initial segment of T (ω), i.e. Te(ω) =
T (ω) ∩ {1, . . . , Nω } for some Nω ∈ N.

After these theorems about partition structures on an abstract metric space,
we return to our more geometric setting of a Gromov triple (X, o, b):
Proposition 9.1.9 (Cf. [73, Lemma 5.13], Footnote 1). Let G  Isom(X) be
nonelementary. Then for all σ > 0 sufficiently large and for every 0 < s < δeG ,

there exist τ > 0, a tree T on N, and an embedding T ∋ ω 7→ xω ∈ G(o) such that
if
Pω := Shad(xω , σ),

then (Pω )ω∈T is an s-thick partition structure on (∂X, D), whose limit set is a
subset of Λur,τ ∩ Λr,σ .
Proof of Theorem 1.2.3 using Theorem 9.1.8 & Proposition 9.1.9.
We first demonstrate the “moreover” clause. Fix σ > 0 large enough such that
e and let (Pω )ω∈T be the partition structure
Proposition 9.1.9 holds. Fix 0 < s < δ,

guaranteed by Proposition 9.1.9. Since this structure is s-thick, applying Theorem
9.1.8 yields a substructure (Pω )ω∈Te whose limit set Js ⊆ Λur,τ ∩ Λr,σ is Ahlfors
s-regular, where τ > 0 is as in Proposition 9.1.9. Since 0 < s < δe was arbitrary,

this completes the proof of the “moreover” clause.

To demonstrate (1.2.2), note that the inequality dimH (Λr ) ≤ δe has already
been established (Lemma 9.0.9), and that the inequalities
dimH (Λur ∩ Λr,σ ) ≤ dimH (Λur ) ≤ dimH (Λr )
are obvious. Thus it suffices to show that
e
dimH (Λur ∩ Λr,σ ) ≥ δ.

But the mass distribution principle guarantees that

dimH (Λur ∩ Λr,σ ) ≥ dimH (Js ) ≥ s
e This completes the proof.
for each 0 < s < δ.



9.1. PARTITION STRUCTURES

139

Proof of Theorem 9.1.8. We will recursively define a sequence of maps
µn : T (n) → [0, 1]
with the following consistency property:
X
µn+1 (ωa).
(9.1.4)
µn (ω) =
a∈T (ω)

The Kolmogorov consistency theorem will then guarantee the existence of a measure
µ
e on T (∞) satisfying
µ
e([ω]) = µn (ω)

(9.1.5)

for each ω ∈ T (n).

Let c = 1 − λs > 0, where λ is as in (9.1.2). For each n ∈ N, we will demand
of our function µn the following property: for all ω ∈ T (n), if µn (ω) > 0, then
cDωs ≤ µn (ω) < Dωs .

(9.1.6)

s
We now begin our recursion. For the case n = 0, let µ0 () := cD∅
; (9.1.6) is

clearly satisfied.
For the inductive step, fix n ∈ N and suppose that µn has been constructed

satisfying (9.1.6). Fix ω ∈ T (n), and suppose that µn (ω) > 0. Formulas (9.1.3)
and (9.1.6) imply that
X
s
Dωa
> µn (ω).
a∈T (ω)

Let Nω ∈ T (ω) be the smallest integer such that
X
s
(9.1.7)
Dωa
> µn (ω).2
a≤Nω

Then the minimality of Nω says precisely that
X
s
Dωa
≤ µn (ω).
a≤Nω −1

Using the above, (9.1.7), and (9.1.2), we have
X
s
s
(9.1.8)
µn (ω) <
≤ µn (ω) + λs Dωs .
Dωa
≤ µn (ω) + DωN
ω
a≤Nω

For each a ∈ T (ω) with a > Nω , let µn+1 (ωa) = 0, and for each a ≤ Nω , let
Ds µn (ω)
µn+1 (ωa) = P ωa
s .
b≤Nω Dωb

Obviously, µn+1 defined in this way satisfies (9.1.4). Let us prove that (9.1.6)
holds (of course, with n = n + 1). The second inequality follows directly from the
2Obviously, this and similar sums are restricted to T (ω).

140

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

definition of µn+1 and from (9.1.7). Using (9.1.8), (9.1.6) (with n = n), and the
equation c = 1 − λs , we deduce the first inequality as follows:


s
λs Dωs
Dωa
µn (ω)
s
= Dωa 1 −
µn+1 (ωa) ≥
µn (ω) + λs Dωs
µn (ω) + λs Dωs


s
λ
s
≥ Dωa
1−
c + λs
s
= cDωa
.

The proof of (9.1.6) (with n = n + 1) is complete. This completes the recursive
step.
Let
Te =

∞
[

{ω ∈ T (n) : µn (ω) > 0}.

n=1

Clearly, the limit set of the partition structure (Pω )ω∈Te is exactly the topological
support of µ := π[e
µ], where µ
e is defined by (9.1.5). Furthermore, for each ω ∈ Te,
we have Te(ω) = T (ω) ∩ {1, . . . , Nω }. Thus, to complete the proof of Theorem 9.1.8
it suffices to show that the measure µ is Ahlfors s-regular.
To this end, fix z = π(ω) ∈ Supp(µ) and 0 < r ≤ κD∅ , where κ is as in (9.1.1)

and (9.1.2). For convenience of notation let

Pn := Pω1n , Dn := Diam(Pn ),
and let n ∈ N be the largest integer such that r < κDn . We have
(9.1.9)

κ2 Dn ≤ κDn+1 ≤ r < κDn .

(The first inequality comes from (9.1.2), whereas the latter two come from the
definition of r.)
We now claim that
B(z, r) ⊆ Pn .
Indeed, by contradiction suppose that w ∈ B(z, r) \ Pn . By (9.1.1) we have
D(z, w) ≥ D(z, Z \ Pn ) ≥ κDn > r
which contradicts the fact that w ∈ B(z, r).

Let k ∈ N be large enough so that λk ≤ κ2 . It follows from (9.1.9) and repeated
applications of the second inequality of (9.1.2) that
Dn+k ≤ λk Dn ≤ κ2 Dn ≤ r,
and thus
Pn+k ⊆ B(z, r) ⊆ Pn .

9.2. A PARTITION STRUCTURE ON ∂X

141

Thus, invoking (9.1.6), we get
(9.1.10)

s
(1 − λs )Dn+k
≤ µ(Pn+k ) ≤ µ(B(z, r)) ≤ µ(Pn ) ≤ Dns .

On the other hand, it follows from (9.1.9) and repeated applications of the first
inequality of (9.1.2) that
Dn+k ≥ κk Dn ≥ κk−1 r.

(9.1.11)

Combining (9.1.9), (9.1.10), and (9.1.11) yields
(1 − λs )κs(k−1) rs ≤ µ(B(z, r)) ≤ κ−2s rs ,
i.e. µ is Ahlfors s-regular. This completes the proof of Theorem 9.1.8.



9.2. A partition structure on ∂X
We begin by stating our key lemma.
Lemma 9.2.1 (Construction of children; cf. [73, Lemma 5.14], Footnote 1). Let
G  Isom(X) be nonelementary. Then for all σ > 0 sufficiently large, for every
0 < s < δeG , for every 0 < λ < 1, and for every w ∈ G(o), there exists a finite

subset T (w) ⊆ G(o) (the children of w) such that if we let
Px := Shad(x, σ)
Dx := Diam(Px )
then the following hold:

(i) The family (Px )x∈T (w) consists of pairwise disjoint shadows contained in
Pw .
(ii) There exists κ > 0 independent of w such that for all x ∈ T (w),
D(Px , ∂X \ Pw ) ≥ κDw
κDw ≤ Dx ≤ λDw .
(iii)

X

x∈T (w)

s
Dxs ≥ Dw
.

It is not too hard to deduce Proposition 9.1.9 from Lemma 9.2.1. We do it
now:
Proof of Proposition 9.1.9 assuming Lemma 9.2.1. Let σ > 0 be large
e and let λ = 1/2. For each
enough so that Lemma 9.2.1 holds. Fix 0 < s < δ,
N (w)

w ∈ G(o), let (yn (w))n=1 be an enumeration of T (w). Define a tree T ⊆ N∗ and

142

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

o

o
g

−1

g(o)

g−1 (o)

Figure 9.2.1. The strategy for the proof of Lemma 9.2.1. To
construct a collection of “children” of the point w = g(o), we “pull
back” the entire picture via g −1 . In the pulled-back picture, the
Big Shadows Lemma 4.5.7 guarantees the existence of many points
x ∈ G(o) such that Shadz (x, σ) ⊆ Shadz (o, σ), where z = g −1 (o).
(Cf. Lemma 9.2.5 below.) These children can then be pushed
forward via g to get children of w.
a collection (xω )ω∈T inductively as follows:
x∅ = o
T (ω) = {1, . . . , N (xω )}
xωa = ya (xω ).
Then the conclusion of Lemma 9.2.1 precisely implies that (Pω := Pxω )ω∈T is an
s-thick partition structure on (∂X, D).
To complete the proof, we must show that the limit set of the partition structure
(Pω )ω∈T is contained in Λur,τ ∩ Λr,σ for some τ > 0. Indeed, fix ω ∈ T (∞). Then

for each n ∈ N, π(ω) ∈ Pω1n = Shad(xω1n , σ) and kxω1n k → ∞. So, the sequence
(xω1n )∞
1 converges σ-radially to π(ω). On the other hand,
d(xω1n , xωn+1 ) ≍+,σ B o (xωn+1 , xω1n )
1

1

(by Lemma 4.5.8)

− logb (κ) ≍+,κ 0.

(by (9.1.2))

≍+,σ − logb
≤

(by (4.5.2))
!

Dωn+1
1

Dω1n

Thus the sequence (xω1n )∞
1 converges to π(ω) τ -uniformly radially, where τ depends
only on σ and κ (which in turn depends on s).

The proof of Lemma 9.2.1 will proceed through a series of lemmas.
Lemma 9.2.2 (Cf. [73, Lemma 5.15]). Fix τ > 0, and let Sτ ⊆ G(o) be a

maximal τ -separated subset. Let B ⊆ bord X be an open set which intersects Λ.

9.2. A PARTITION STRUCTURE ON ∂X

C−3 C−2 C−1 C0

C1

143

C2

g−

g+

Figure 9.2.2. The sets Cn , for n ∈ Z.

e the series
Then for every 0 < t < δ,

Σt (Sτ ∩ B)

diverges.
Proof. By Proposition 7.4.6, there exists a loxodromic isometry g ∈ G such

that g+ ∈ B. Let ℓ(g) = logb g ′ (g− ) = − logb g ′ (g+ ) > 0, and let the functions

r = rg+ ,g− ,o , θ = θg+ ,g− ,o be as in Section 4.6. Fix N ∈ N large to be determined,
let κ = N ℓ(g), and for each n ∈ Z let
Cn = {x ∈ X : nκ ≤ r(x) < (n + 1)κ}
(cf. Figure 9.2.2). Let
C+,0 =

[

Cn ,

C+,1 =

n≥0
even

C−,0 =

[

[

Cn

n≥0
odd

Cn ,

C−,1 =

n<0
even

[

Cn .

n<0
odd

Fix ρ > 0, and let Sρ ⊆ G(o) be a maximal ρ-separated set. Since Σt (Sρ ) = ∞,

one of the series Σt (Sρ ∩ C+,0 ), Σt (Sρ ∩ C+,1 ), Σt (Sρ ∩ C−,0 ), and Σt (Sρ ∩ C−,1 )

must diverge. By way of illustration let us consider the case where
Σt (Sρ ∩ C−,0 ) = ∞.

144

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

Let
Aρ =

[

n<0
even

g 2N |n| (Cn ∩ Sρ ).

Claim 9.2.3. Σt (Aρ ) = ∞.
Proof. Fix n = −m < 0 even and x ∈ Cn . Then by (4.6.1),
r(g 2N m (x)) ≍+ 2N mℓ(g) + r(x) = 2mκ + r(x) ≍+,κ 2mκ − mκ = mκ
and thus
|r(g 2N m (x))| ≍+,κ |r(x)|.

On the other hand, by (4.6.2) we have θ(g 2N m (x)) ≍+ θ(x). Combining with

Lemma 4.6.2 gives

d(0, x) ≍+,κ d(0, g 2N m (x)).
Thus
Σt (Aρ ) =

X

X

b−tkg

2N m

(x)k

m>0 x∈C−m ∩Sρ
even

≍×,κ

X

X

b−tkxk

m>0 x∈C−m ∩Sρ
even

= Σt (C−,0 ∩ Sρ ) = ∞.
⊳
Claim 9.2.4. Aρ is a ρ-separated set.
Proof. Fix y1 , y2 ∈ Aρ . Then for some m1 , m2 > 0 even, we have xi :=

g −2N mi (yi ) ∈ C−mi (i = 1, 2). If n1 = n2 , then we have
d(y1 , y2 ) = d(x1 , x2 ) ≥ ρ,

since x1 , x2 ∈ Sρ and Sρ is ρ-separated. So suppose n1 6= n2 ; without loss of
generality we may assume n1 > n2 . Then by (4.6.1) we have
r(y1 ) − r(y2 ) ≍+ 2N m1 ℓ(g) + r(x1 ) − (2N m2 ℓ(g) + r(x2 ))
= 2κ(m1 − m2 ) + r(x1 ) − r(x2 )
≥ 2κ[m1 − m2 ] + κ(−m1 ) − κ(−m2 + 1)
= κ(m1 − m2 − 1)
≥ κ = N ℓ(g).
By choosing N sufficiently large, we may guarantee that r(y1 ) − r(y2 ) ≥ ρ, which
implies d(y1 , y2 ) ≥ ρ.

⊳

9.2. A PARTITION STRUCTURE ON ∂X

145

For all x ∈ Nρ (Aρ ), we have r(x) &+ 0. Thus g− ∈
/ Nρ (Aρ ). So by Theorem

6.1.10, we can find n ∈ N such that Nρ (g n (Aρ )) ⊆ B.
Let Sρ/2 ⊆ G(o) be a maximal ρ/2-separated set. By Lemma 8.2.6, we have
Σt (Sρ/2 ∩ B) &× Σt (g n (Aρ )) ≍× Σt (Aρ ) = ∞.
Since ρ > 0 was arbitrary, this completes the proof.



Lemma 9.2.5 (Cf. [73, Sublemma 5.17], Footnote 1). Let B ⊆ bord X be an
e
open set which intersects Λ. For all σ > 0 sufficiently large and for all 0 < s < δ,

there exists a set SB ⊆ G(o) ∩ B such that for all z ∈ X \ B,
(i) If

Pz,x := Shadz (x, σ),
then the family (Pz,x )x∈SB consists of pairwise disjoint shadows contained
in Pz,o ∩ B.

(ii) There exists κ > 0 independent of z (but depending on s) such that for all
x ∈ SB ,
(9.2.1)

Db,z (Pz,x , ∂X \ Pz,o ) ≥ κ Diamz (Pz,o )

(9.2.2)

κ Diamz (Pz,o ) ≤ Diamz (Pz,x ) ≤ λ Diamz (Pz,o ).

(iii)

X

x∈SB

Diamsz (Pz,x ) ≥ Diamsz (Pz,o ).

Proof. Let B ⊆ bord X be an open set which contains a point η ∈ Λ. Choose
ρ > 0 large enough so that
{x ∈ bord X : hx|ηio ≥ ρ} ⊆ B.
Then fix σ > 0 large to be determined, depending only on ρ. Fix ρe ≥ ρ large to be
determined, depending only on ρ and σ.
Fix 0 < s < δe and z ∈ X \ B. For all x ∈ X we have
0 ≍+,ρ hz|ηio &+ min(hx|ηio , hx|zio ).

Let
e = {x ∈ X : hx|ηio ≥ ρe}.
B

If ρe is chosen large enough, then we have

hx|zio ≍+,ρ 0,

(9.2.3)

e We emphasize that the implied constants of these asymptotics are
for all x ∈ B.

independent of both z and s.
For each n ∈ N let

An := B(o, n) \ B(o, n − 1)

146

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

be the nth annulus centered at o. We shall need the following variant of the Intersecting Shadows Lemma:
Claim 9.2.6. There exists τ > 0 depending on ρ and σ such that for all n ∈ N

e if
and for all x, y ∈ An ∩ B,

Pz,x ∩ Pz,y 6= ,

then

d(x, y) < τ.
Proof. Without loss of generality suppose d(z, y) ≥ d(z, x). Then by the
Intersecting Shadows Lemma 4.5.4 we have
d(x, y) ≍+,σ B z (y, x) = B o (y, x) + 2hx|zio − 2hy|zio .
e we have
Now | B o (y, x)| ≤ 1 since x, y ∈ An . On the other hand, since x, y ∈ B,
hx|zio ≍+,ρ hy|zio ≍+,ρ 0.

Combining gives
d(x, y) ≍+,ρ,σ 0,
and letting τ be the implied constant finishes the proof.

⊳

Fix M > 0 large to be determined, depending on ρ and τ (and thus implicitly
e then by Lemma
on σ). Let Sτ ⊆ G(o) be a maximal τ -separated set. Fix t ∈ (s, δ);

9.2.2 we have

e =
∞ = Σt (Sτ ∩ B)
=

∞
X

n=1
∞
X

e ∩ An )
Σt (Sτ ∩ B
X

b−tkxk

n=1 x∈Sτ ∩B∩A
e
n

≍×

∞
X

n=1

b−(t−s)n

X

b−skxk .

e
x∈Sτ ∩B∩A
n

It follows that there exist arbitrarily large numbers n ∈ N such that
X
b−skxk ≥ M.
(9.2.4)
e
x∈Sτ ∩B∩A
n

Fix such an n, also to be determined, depending on λ, ρ, ρe, and M (and thus
e ∩ An . To complete the proof, we
implicitly on τ and σ), and let SB = Sτ ∩ B
demonstrate (i)-(iii).
Proof of (i). In order to see that the shadows (Pz,x )x∈SB are pairwise disjoint, suppose that x, y ∈ SB are such that Pz,x ∩ Pz,y 6= . By Claim 9.2.6 we

have d(x, y) < τ . Since SB is τ -separated, this implies x = y.

9.2. A PARTITION STRUCTURE ON ∂X

147

Fix x ∈ SB . Using (9.2.3) and the fact that x ∈ An , we have
ho|zix ≍+ kxk − hx|zio ≍+,ρ kxk ≍+ n.
Thus for all ξ ∈ Pz,x ,
0 ≍+,σ hz|ξix &+ min(ho|zix , ho|ξix ) ≍+ min(n, ho|ξix );
taking n sufficiently large (depending on σ), this gives
ho|ξix ≍+,σ 0,
from which it follows that
hx|ξio ≍+ d(o, x) − ho|ξix ≍+,σ n.
e we get
Therefore, since x ∈ B,

hξ|ηio &+ min(hx|ξio , hx|ηio ) &+,σ min(n, ρe).

Thus ξ ∈ B as long as ρe and n are large enough (depending on σ). Thus Pz,x ⊆ B.

Finally, note that we do not need to prove that Pz,x ⊆ Pz,o , since it is implied
by (9.2.1) which we prove below.
⊳

Proof of (ii). Take any x ∈ SB . Then by (9.2.3), we have
(9.2.5)

d(x, z) − kzk = kxk − 2hx|zio ≍+,ρ kxk ≍+ n.

Combining with the Diameter of Shadows Lemma 4.5.8 gives
(9.2.6)

b−d(z,x)
Diamz (Pz,x )
≍×,σ −d(z,o) ≍×,ρ b−n .
Diamz (Pz,o )
b

Thus by choosing n sufficiently large depending on σ, λ, and ρ (and satisfying
(9.2.4)), we guarantee that the second inequality of (9.2.2) holds. On the other
hand, once n is chosen, (9.2.6) guarantees that if we choose κ sufficiently small,
then the first inequality of (9.2.2) holds.
In order to prove (9.2.1), let ξ ∈ Pz,x and let γ ∈ ∂X \ Pz,o . We have
hx|ξiz ≍+ d(x, z) − hz|ξix ≥ d(x, z) − σ
ho|γiz ≍+ kzk − ho|ξix ≤ kzk − σ.
Also, by (9.2.3) we have
ho|xiz ≍+ kzk − hx|zio ≍+,ρ kzk.

148

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

Applying Gromov’s inequality twice and then applying (9.2.5) gives
kzk − σ &+ ho|γiz &+ min (ho|xiz , hx|ξiz , hξ|γiz )
&+,ρ min (kzk, d(x, z) − σ, hξ|γiz )
≍+ min (kzk, kzk + n − σ, hξ|γiz ) .
By choosing n and σ sufficiently large (depending on ρ), we can guarantee that neither of the first two expressions can represent the minimum without contradicting
the inequality. Thus
kzk − σ &+,ρ hξ|γiz ;
exponentiating and the Diameter of Shadows Lemma 4.5.8 give
Db,z (ξ, γ) &×,ρ b−(kzk−σ) ≍×,σ b−kzk ≍×,σ Diamz (Pz,o ).
Thus we may choose κ small enough, depending on ρ and σ, so that (9.2.1) holds.
⊳
Proof of (iii).
X
Diamsz (Pz,x ) ≍×

x∈SB

X

b−sd(z,x)

x∈SB

≍×,ρ b−sd(z,o)

≥

X

(by the Diameter of Shadows Lemma)
b−kxk

(by (9.2.5))

x∈SB

M b−sd(z,o)

≍× M Diamsz (Pz,o ).

(by (9.2.4))
(by the Diameter of Shadows Lemma)

Letting M be larger than the implied constant yields the result.

⊳


We may now complete the proof of Lemma 9.2.1:
Proof of Lemma 9.2.1. Let η1 , η2 ∈ Λ be distinct points, and let B1 and B2

be disjoint neighborhoods of η1 and η2 , respectively. Let σ > 0 be large enough so
e and let S1 ⊆ G(o) ∩ B1
that Lemma 9.2.5 holds for both B1 and B2 . Fix 0 < s < δ,
and S2 ⊆ G(o) ∩ B2 be the sets guaranteed by Lemma 9.2.5. Now suppose that

−1
w = gw (o) ∈ G(o). Let z = gw
(o). Then either z ∈
/ B1 or z ∈
/ B2 ; say z ∈
/ Bi . Let
T (w) = gw (Si ); then (i)-(iii) of Lemma 9.2.5 exactly guarantee (i)-(iii) of Lemma

9.2.1.


9.3. Sufficient conditions for Poincaré regularity

We end this chapter by relating the modified Poincaré exponent to the classical
Poincaré exponent under certain additional assumptions, thus completing the proof
of Theorem 1.2.1.

9.3. SUFFICIENT CONDITIONS FOR POINCARÉ REGULARITY

that

149

Proposition 9.3.1. Let G ≤ Isom(X) be nonelementary, and assume either
(1) X is regularly geodesic and G is moderately discrete,
(2) X is an algebraic hyperbolic space and G is weakly discrete, or that
(3) X is an algebraic hyperbolic space and G is COT-discrete and acts irreducibly.

Then G is Poincaré regular.
Remark 9.3.2. Example 13.4.2 shows that Proposition 9.3.1 cannot be improved by replacing “COT” with “UOT”, Example 13.4.9 shows that Proposition
9.3.1 cannot be improved by removing the assumption that G acts irreducibly, Example 13.4.1 shows that Proposition 9.3.1 cannot be improved by removing the
hypothesis that X is an algebraic hyperbolic space from (ii), and Example 13.4.4
shows that Proposition 9.3.1 cannot be improved by removing the assumption that
X is regularly geodesic.
We begin with the following observation:
Observation 9.3.3. If (3) implies that G is Poincaré regular, then (2) does as
well.
Proof. Suppose (2) holds, and let S be the smallest totally geodesic subset of
bord X which contains Λ (cf. Lemma 2.4.5). Since G is nonelementary, V := S ∩ X

is nonempty; it is clear that V is G-invariant. By Observation 5.2.14, the action G ↿
V is weakly discrete. By Proposition 5.2.7, G ↿ V is COT-discrete. Furthermore, G

acts irreducibly on V because of the way V was defined (cf. Proposition 7.6.3). Thus
e G (since the
(3) holds for the action G ↿ V , which by our hypothesis implies ∆G = ∆
Poincaré set and modified Poincaré set are clearly stable under restrictions).



We now proceed to prove that (1) and (3) each imply that G is Poincaré regular.
By contradiction, let us suppose that G is Poincaré irregular. By Proposition
8.2.4(ii), we have that G is not strongly discrete and thus
δeG < δG = ∞.

This gives us two contrasting behaviors: On one hand, by Proposition 8.2.4(iii),
there exist ρ > 0 and a maximal ρ-separated set Sρ ⊆ G(o) so that Sρ does not
contain an bounded infinite set. On the other hand, since G is not strongly discrete,
there exists σ > 0 such that #(Gσ ) = ∞, where
Gσ := {g ∈ G : g(o) ∈ B(o, σ)}.
Claim 9.3.4. For every ξ ∈ Λ, the orbit Gσ (ξ) is precompact.

150

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

Proof. Suppose not. Then the set Gσ (ξ) is complete (with respect to the
metric D) but not compact. It follows that Gσ (ξ), and thus also Gσ (ξ), is not
totally bounded. So there exists ε > 0 and an infinite ε-separated subset (gn (ξ))∞
1 .
Fix L large to be determined. Since ξ ∈ Λ, we can find x ∈ G(o) such that
hx|ξio ≥ L.
Subclaim 9.3.5. By choosing L large enough we can ensure
d(gm (x), gn (x)) ≥ 2ρ ∀m, n ∈ N.
Proof. By (d) of Proposition 3.3.3,
hgn (x)|gn (ξ)io ≍+,σ hgn (x)|gn (ξ)ign (o) = hx|ξio ≥ L,
and thus
D(gn (x), gn (ξ)) .×,σ b−L .
If L is large enough, then this implies
D(gn (x), gn (ξ)) ≤ ε/3.
Since by construction the sequence (gn (ξ))∞
1 is ε-separated, we also have
D(gm (ξ), gn (ξ)) ≥ ε
and then the triangle inequality gives
D(gm (x), gn (x)) ≥ ε/3,
or, taking logarithms,
hgm (x)|gn (x)io .+ − logb (ε/3).
Now we also have
kgn (x)k ≍+,σ kxk ≥ hx|ξio ≥ L
and therefore
d(gm (x), gn (x)) =

kgm (x)k + kgn (x)k − 2hgm (x)|gn (x)io

&+,σ 2L − 2(− logb (ε/3)).
Thus by choosing L sufficiently large, we ensure that d(gm (x), gn (x)) ≥ 2ρ.

⊳

Recall that Sρ is a maximal ρ-separated set. Thus for each n ∈ N, we can find

yn ∈ Sρ with d(gn (x), yn ) < ρ. Then the subclaim implies yn 6= ym for n 6= m. But
on the other hand
kyn k ≤ kxk + σ + ρ ∀n ∈ N,
which implies that Sρ contains a bounded infinite set, a contradiction.



9.3. SUFFICIENT CONDITIONS FOR POINCARÉ REGULARITY

151

We now proceed to disprove the hypotheses (1) and (3) of Proposition 9.3.1.
Thus if either of these hypotheses are assumed, we have a contradiction which
finishes the proof.

Proof that (1) cannot hold. Since G is assumed to be nonelementary, we
can find distinct points ξ1 , ξ2 ∈ Λ. By Claim 9.3.4, there exist a sequence (gn )∞
1 in
Gσ and points η1 , η2 ∈ Λ such that

gn (ξi ) −
→ ηi .
n

Next, choose a point x ∈ [ξ1 , ξ2 ]. For each n ∈ N, we have
gn (x) ∈ [gn (ξ1 ), gn (ξ2 )].
Thus since X is regularly geodesic there exist a sequence (nk )∞
1 and a point z ∈

[η1 , η2 ] such that

gnk (x) −
→ z.
k

Since gn ∈ Gσ

∀n, the sequence (gn (x))∞
1 is bounded and thus z ∈ X. By

contradiction, suppose that G is moderately discrete, and fix ε > 0 satisfying

(5.2.2). For all m, n ∈ N large enough so that gm (x), gn (x) ∈ B(z, ε/2), we have
−1
gn (x)) = d(gm (x), gn (x)) ≤ ε. Thus for some N ∈ N, we have
d(x, gm
−1
#{gm
gn : m, n ≥ N } < ∞.

This is clearly a contradiction.



Proof that (3) cannot hold. Now we assume that X is an algebraic hyperbolic space, say X = H = Hα
F , and that G acts irreducibly on X. Using the
identification
Isom(H) ≡ O∗ (L; Q)/ ∼,

(Theorem 2.3.3), for each g ∈ Gσ let Tg ∈ O∗ (L; Q) be a representative of g. Recall

(Lemma 2.4.11) that

kTg k = kTg−1k = ekgk ,

so since g ∈ Gσ we have kTg k = kTg−1 k ≤ bσ . In particular, the family (Tg )g∈Gσ
acts equicontinuously on L.

For simplicity of exposition, in the following proof we will assume that X

is separable. (In the non-separable case, the reader should use nets instead of
sequences.) It follows that Λ ⊆ ∂X is also separable; let (ξk = [xk ])∞
1 be a dense
sequence, with xk ∈ L, kxk k = 1.

152

9. GENERALIZATION OF THE BISHOP–JONES THEOREM

Claim 9.3.6. There exists a sequence of distinct elements (gn )∞
1 in Gσ such
that the following hold:
(+)

∈ L \ {0}

(−)

∈ L \ {0}

→ yk
Tgn [xk ] −
n

[xk ] −
→ yk
Tg−1
n
n

→ σ ∈ Aut(F).
σ(Tgn ) −
n

Proof. For each k ∈ N let
Kk = {y ∈ L \ {0} : [y] ∈ Gσ (ξk ) and b−σ ≤ kyk ≤ bσ },
and let
K :=

Y

k∈N

Kk

!2

× Aut(F).

Then by Claim 9.3.4 (and general topology), K is a compact metrizable space, and
is in particular sequentially compact. Now for each g ∈ Gσ ,

b−σ ≤ kTg [xk ]k ≤ bσ and b−σ ≤ kTg−1 [xk ]k ≤ bσ
and thus


−1
∞
φg := (Tg (xk ))∞
1 , (Tg (xk ))1 , σ(Tg ) ∈ K,

and so since #(Gσ ) = ∞, there exists a sequence of distinct elements (gn )∞
1 in Gσ
so that the sequence (φgn )∞
1 converges to a point


(+)
(−) ∞
(yk )∞
1 , (yk )1 , σ ∈ K.
Writing out what this means yields the claim.

⊳

Let Tn = Tgn and σn = σ(Tn ) → σ. We claim that the sequence (Tn )∞
1 is
convergent in the strong operator topology. Let
K = {a ∈ F : σ(a) = a}

V = {x ∈ L : the sequence (Tn [x])∞
1 converges}.
Then K is an R-subalgebra of F, and V is a K-module. Given x, y ∈ V , by Observation 2.3.6 we have
σn (BQ (x, y)) = BQ (Tn x, Tn y) −
→ BQ (x, y),
n

so B(x, y) ∈ K. Thus V satisfies (2.4.1). On the other hand, since the family

(Tn )∞
1 acts equicontinuously on L, the set V is closed. Thus [V ] ∩ bord X is totally
geodesic. But by construction, ξk ∈ [V ] for all k, and so
Λ ⊆ [V ]

9.3. SUFFICIENT CONDITIONS FOR POINCARÉ REGULARITY

153

Therefore, since by hypothesis G acts irreducibly, it follows that [V ] = X, i.e.
V = L (Proposition 7.6.3). So for every x ∈ L, the sequence (Tn (x))∞
1 converges.
Thus
Tn −
→ T (+) ∈ L(L)
n

in the strong operator topology. (The boundedness of the operator T (+) follows
from the uniform boundedness of the operators (Tn )∞
1 .) We do not yet know that
(+)
T
is invertible. But a similar argument yields that
Tn−1 −
→ T (−) ∈ L(L),
n

and since the sequences

(Tn )∞
1

and (Tn−1 )∞
1 are equicontinuous, we have

T (+) T (−) = lim Tn Tn−1 = I,
n→∞

and similarly T

(−)

T

(+)

= I. Thus T

(+)

and T (−) are inverses of each other and in

particular
T (+) ∈ O∗ (L; Q).

Let h = [T (+) ] ∈ Isom(X). By Proposition 5.1.2, we have gn → h in the compact-

open topology. Thus, Lemma 5.2.8 completes the proof.



Part 3

Examples

This part will be divided as follows: In Chapter 10 we consider semigroups of
isometries which can be written as the “Schottky product” of two subsemigroups.
Next, we analyze in detail the class of parabolic groups of isometries in Chapter
11. In Chapter 12, we define a subclass of the class of groups of isometries which
we call geometrically finite, generalizing known results from the Standard Case.
In Chapter 13, we provide a list of examples whose main importance is that they
are counterexamples to certain implications; however, these examples are often
geometrically interesting in their own right. Finally, in Chapter 14, we consider
methods of constructing R-trees which admit natural group actions, including what
we call the “stapling method”.

CHAPTER 10

Schottky products
An important tool for constructing examples of discrete subgroups of Isom(X)
is the technique of Schottky products. Schottky groups are a special case of Schottky
products; cf. Definition 10.2.4. In this chapter we explain the basics of Schottky
products on hyperbolic metric spaces, and give several important examples. We
intend to give a more comprehensive account of Schottky products in [57], where
we will study their relation to pseudo-Markov systems (defined in [159]).
Remark 10.0.7. Throughout this chapter, E denotes an index set with at least
two elements. There are no other restrictions on the cardinality of E; in particular,
E may be infinite.
10.1. Free products
We provide a brief review of the theory of free products, mainly to fix notation.
Let (Γa )a∈E be a collection of nontrivial abstract semigroups. Let
a
[
ΓE =
(Γa \ {e}) =
{a} × (Γa \ {e}).
a∈E

a∈E

∗

Let (ΓE ) denote the set of finite words with letters in ΓE , including the empty
word, which we denote by . The free product of (Γa )a∈E , denoted ∗a∈E Γa , is the
set
{g = (a1 , γ1 ) · · · (an , γn ) ∈ (ΓE )∗ : ai 6= ai+1 ∀i = 1, . . . , n − 1, n ≥ 0} ,
together with the operation of multiplication defined as follows: To multiply two
words g, h ∈ ∗a∈E Γa , begin by concatenating them. The concatenation may no
longer satisfy ai 6= ai+1 for all i; namely, this condition may fail at the point where

the two words are joined. Reduce the concatenated word g ∗ h using the rule

(a, γ γ ) γ γ 6= e
1 2
1 2
.
(a, γ1 )(a, γ2 ) =

γ γ =e
1 2

The word may require multiple reductions in order to satisfy ai 6= ai+1 . The

reduced form of g ∗ h will be denoted gh.
One verifies that the operation of multiplication defined above is associative, so
that the free product ∗a∈E Γa is a semigroup. If (Γa )a∈E are groups, then ∗a∈E Γa
157

158

10. SCHOTTKY PRODUCTS

is a group. The inclusion maps ιa : Γa → ∗a∈E Γa defined by ιa (γ) = (a, γ) are

homomorphisms, and ∗a∈E Γa = hιa (Γa )ia∈E .
An important fact about free products is their universal property: Given any

semigroup Γ and any collection of homomorphisms (πa : Γa → Γ), there exists a
unique homomorphism π : ∗a∈E Γa → Γ such that πa = π ◦ιa for all a. For example,

if (Γa )a∈E are subsemigroups of Γ and (πa )a∈E are the identity inclusions, then
π((a1 , γ1 ) · · · (an , γn )) = γ1 · · · γn . We will call the map π the natural map from

∗a∈E Γa to Γ.

Remark 10.1.1. We will use the notation Γ1 ∗ · · · ∗ Γn to denote ∗a∈{1,...,n} Γa .

The semigroups

· · ∗ N}
Fn (Z) = Z
· · ∗ Z} and Fn (N) = |N ∗ ·{z
| ∗ ·{z
n times

n times

are called the free group on n elements and the free semigroup on n elements,
respectively.
10.2. Schottky products
Given a collection of semigroups Ga  Isom(X), we can ask whether the semi-

group hGa ia∈E  Isom(X) is isomorphic to the free product ∗a∈E Ga . A sufficient
condition for this is that the groups (Ga )a∈E are in Schottky position.
Definition 10.2.1. A collection of nontrivial semigroups (Ga  Isom(X))a∈E
is in Schottky position if there exist disjoint open sets Ua ⊆ bord X satisfying:
(I) For all a, b ∈ E distinct and g ∈ Ga \ {id},

(II) There exists o ∈ X \
(10.2.1)

S

g(Ub ) ⊆ Ua .

a∈E

Ua satisfying

g(o) ∈ Ua ∀a ∈ E ∀g ∈ Ga \ {id}.

Such a collection (Ua )a∈E is called a Schottky system for (Ga )a∈E . If the collection
(Ga )a∈E is in Schottky position, then we will call the semigroup G = hGa ia∈E the
Schottky product of (Ga )a∈E .
A Schottky system will be called global if for all a ∈ E and g ∈ Ga \ {id},
(10.2.2)

g(bord X \ Ua ) ⊆ Ua .

Remark. In most references (e.g. [56, §5]), (10.2.2) or a similar hypothesis

is taken as the definition of Schottky position. So what these references call a

“Schottky group”, we would call a “global Schottky group”. There are important
examples of Schottky semigroups which are not global; see e.g. (B) of Proposition
10.5.4. It should be noted that such examples tend to be semigroups rather than

10.2. SCHOTTKY PRODUCTS

159

groups, which explains why references which consider only groups can afford to
include globalness in their definition of Schottky position.
Remark. The above definition may be slightly confusing to someone familiar
with classical Schottky groups, since in that context the sets Ua in the above definition are not half-spaces but rather unions of pairs of half-spaces; cf. Definition
10.2.4.
The basic properties of Schottky products are summarized in the following
lemma:
Lemma 10.2.2. Let G = hGa ia∈E be a Schottky product. Then:

(i) (Ping-Pong Lemma) The natural map π : ∗a∈E Ga → G is an injection
(and therefore an isomorphism).

(ii) Fix g = (a1 , g1 )(a2 , g2 ) · · · (an , gn ) ∈ ∗a∈E Ga , and let g = π(g). Then
g(o) ∈ Ua1 .

(10.2.3)
Moreover, for all b 6= an

g(Ub ) ⊆ Ua1 ,

(10.2.4)

and if the system (Ua )a∈E is global
(10.2.5)

g(bord X \ Uan ) ⊆ Ua1 .

(iii) If G is a group, then G is COT-discrete.
Proof. (10.2.3)-(10.2.5) may be proven by an easy induction argument. Now
(10.2.3) immediately demonstrates (i), since it implies that π(g) 6= id. (iii) also
follows from (10.2.3), since it shows that kgk is bounded from below for all g ∈
G \ {id}.



Remark 10.2.3. Lemma 10.2.2(i) says that Schottky products are (isomorphic
to) free products. However, we warn the reader that the converse is not necessarily
true; cf. Lemma 13.4.6.
Two important classes of Schottky products are Schottky groups and Schottky
semigroups.
Definition 10.2.4. A Schottky group is the Schottky product of cyclic groups
Ga = (ga )Z with the following property: For each a ∈ E, Ua may be written as the
disjoint union of two sets Ua+ and Ua− satisfying

ga (bord X \ Ua− ) ⊆ Ua+ .
A Schottky semigroup is simply the Schottky product of cyclic semigroups; no
additional hypotheses are needed.

160

10. SCHOTTKY PRODUCTS

Remark 10.2.5. In the classical theory of Schottky groups, the sets Ua± are
required to be half-spaces. A half-space in bord Hα is a connected component of the
complement of a totally geodesic subset of bord Hα of codimension one. Requiring
the sets Ua± to be half-spaces has interesting effects on the geometry of Schottky
groups.
Although the notion of half-spaces cannot be generalized to hyperbolic metric
spaces in general or even to nonreal algebraic hyperbolic spaces (since a totally
geodesic subspace of an algebraic hyperbolic space over F = C or Q always has real
codimension at least 2, so deleting it yields a connected set), it at least makes sense
over real hyperbolic spaces and in the context of R-trees. A half-space in an R-tree
X is a connected component of the complement of a point in X.
We hope to study the effect of requiring the sets Ua± to be half-spaces, both
in the case of real (but infinite-dimensional) algebraic hyperbolic spaces and in the
case of R-trees, in more detail in [57].
10.3. Strongly separated Schottky products
Many questions about Schottky products cannot be answered without some
additional information. For example, one can ask whether or not the Schottky
product of strongly (resp. moderately, weakly) discrete groups is strongly (resp.
moderately, weakly) discrete. One can also ask about the relation between the
Poincaré exponent of a Schottky group and the Poincaré exponent of its factors.
For the purposes of this monograph, we will be interested in Schottky products
which satisfy the following condition:
Definition 10.3.1. A Schottky product G = hGa ia∈E is said to be strongly
separated (with respect to a Schottky system (Ua )a∈E ) if there exists ε > 0 such
that for all a, b ∈ E distinct and g ∈ Ga \ {id},
(10.3.1)


D Ua ∪ g −1 (bord X \ Ua ), Ub ≥ ε.

Here D is as in Proposition 3.6.13. Abusing terminology, we will also call the
semigroup G and the Schottky system (Ua )a∈E strongly separated.
The product G = hGa ia∈E is weakly separated if (10.3.1) holds for a constant
ε > 0 which depends on a and b (but not on g).
Remark 10.3.2. There are many important examples of Schottky products
which are not strongly separated, and we hope to analyze these in more detail in
[57]. Some examples of Schottky products that do satisfy the condition are given
in Section 10.5.
Standing Assumptions 10.3.3. For the remainder of this chapter,
G = hGa ia∈E

10.3. STRONGLY SEPARATED SCHOTTKY PRODUCTS

161

denotes a strongly separated Schottky product and (Ua )a∈E denotes the corresponding Schottky system. Moreover, from now on we assume that the hyperbolic
metric space X is geodesic.
Notation 10.3.4. Let Γ denote the free product Γ = ∗a∈E Ga , and let π : Γ →

G denote the natural isomorphism. Whenever we have specified an element g ∈ Γ,

we denote its length by |g| and we write g = (a1 , g1 ) · · · (a|g| , g|g|). For h ∈ Γ, we
write h = (b1 , h1 ) · · · (b|h| , h|h| ).
S
Let o ∈ X satisfy (10.2.1). Let ε ≤ d(o, a Ua ) satisfy (10.3.1), and for each a ∈
E let Va denote the closed ε/4-thickening of Ua with respect to the D metric. Then
the sets (Int(Va ))a∈E are also a Schottky system for (Ga )a∈E ; they are strongly
separated with ε = ε/2; moreover,
D(Ua , bord X \ Va ) ≥ ε/2 ∀a ∈ E.

(10.3.2)
Finally, let

so that


bord X \ Int(V ) (U )
a
a a∈E is global
Xa =
,
{o} ∪ S
otherwise
b6=a Vb
g(Xa ) ⊆ Ua ∀a ∈ E.

(10.3.3)

Note that since the sets (Va )a∈E are ε/2-separated, they have no accumulation
points and thus Xa is closed for all a ∈ E.
The strong separation condition will allow us to relate the discreteness of the
groups Ga to the discreteness of their Schottky product G. It will also allow us
to relate the Poincaré exponents of Ga with the Poincaré exponent of G. The
underlying fact which will allow us to prove both of these relations is the following
lemma:
Lemma 10.3.5. There exist constants C, ε > 0 such that for all g ∈ Γ,
(10.3.4)

|g|
X
i=1

In particular
(10.3.5)

(kgi k − C) ∨ ε ≤ d(X \ Va1 , π(g)(Xa|g| )) ≤

|g|
|g|
X
X
kgi k
(kgi k − C) ∨ ε ≤ kπ(g)k ≤
i=1

i=1

and thus
(10.3.6)

kπ(g)k ≍×

|g|
X
i=1

1 ∨ kgi k

|g|
X
i=1

kgi k.

162

10. SCHOTTKY PRODUCTS

Va
Ua
g2 (Xb )
x1

Ub
Vb

g1 (Xa )

x2

π(g)(o)
o
Vc

x3

Uc

g3 (Xc )

Figure 10.3.1. The geodesic segment [o, π(g)(o]) splits up naturally into four subsegments, which can then be rearranged by the
isometry group to form geodesic segments which connect o with
g1 (Xa ), Va with g2 (Xb ), Vb with g3 (Xc ), and Vc with o, respectively. Here g = (a, g1 )(b, g2 )(c, g3 ).

Proof. The second inequality of (10.3.4) is immediate from the triangle inequality. For the first inequality, fix g ∈ Γ, x ∈ X \ Va1 , and y ∈ Xa|g| . Write
n = |g|. We have
π(g)(y) ∈ g1 · · · gn (Xan ) ⊆ g1 · · · gn−1 (Van ) ⊆ g1 · · · gn−1 (Xan−1 )
⊆ · · · ⊆ g1 (Va2 ) ⊆ g1 (Xa1 ) ⊆ Va1 6∋ x.
Consequently, the geodesic [x, π(g)(y)] intersects the sets
∂Va1 , g1 (∂Xa1 ), g1 (∂Va2 ), . . . , g1 · · · gn−1 (∂Van ), g1 · · · gn (∂Xan )
in their respective orders. Thus
d(x, π(g)(y)) ≥
(10.3.7)

=
≥

n
X

i=1
n
X

i=1
n
X
i=1

d(g1 · · · gi−1 (∂Vai ), g1 · · · gi (∂Xai ))
d(∂Vai , gi (∂Xai ))
d(X \ Vai , gi (Xai )).

10.3. STRONGLY SEPARATED SCHOTTKY PRODUCTS

163

(Cf. Figure 10.3.1.) Now fix i = 1, . . . , n, and we will estimate the distance d(X \
Vai , gi (Xai )). For convenience of notation write a = ai and g = gi .
Fix z ∈ X \ Va and w ∈ g(Xa ). Combining (10.3.2) and (10.3.3) gives
D(z, w) ≥ ε/2
and in particular
d(z, w) ≥ ε/2.

(10.3.8)

On the other hand, converting the inequality D(z, w) ≥ ε/2 into a statement about

Gromov products shows that

d(z, w) ≍+,ε d(o, z) + d(o, w) ≥ d(o, w) ≥ d(g −1 (o), Xa ).
Since D(g −1 (o), Xa ) ≥ D(g −1 (bord X \ Va ), Xa ) ≥ ε/2, we have
d(g −1 (o), Xa ) &+,ε d(g −1 (o), o) = kgk.
Combining with (10.3.8) gives
d(z, w) ≥ (kgk − C) ∨ (ε/2)
for some C > 0 depending only on ε. Taking the infimum over all z, w gives
d(X \ Vai , gi (Xai )) ≥ (kgi k − C) ∨ (ε/2).
Summing over all i = 1, . . . , n and combining with (10.3.7) yields (10.3.4). Since o ∈
Xa|g| and o ∈ X \ Va1 , (10.3.5) follows immediately. Finally, the coarse asymptotic
(kgi k − C) ∨ ε ≍×,C,ε 1 ∨ kgi k.
implies (10.3.6).



Corollary 10.3.6. Suppose that #{a ∈ E : d(o, Ua ) ≤ ρ} < ∞ for all ρ > 0.
If the groups (Ga )a∈E are strongly discrete, then G is strongly discrete.
In fact, this corollary holds even if G is only weakly separated and not strongly
separated.

Proof. Since kgk ≥ d(o, Ua ) for all a ∈ E and g ∈ Ga , our hypotheses implies
that
#{(a, g) ∈ ΓE : kgk ≤ ρ} < ∞ ∀ρ.

164

10. SCHOTTKY PRODUCTS

It follows that for all N ∈ N,


|g|


X
1 ∨ kgi k ≤ N
# g∈Γ:


i=1

(10.3.9)

≤

≤

N
X

n=0
N
X

n=0

# {g ∈ (ΓE )n : kgi k ≤ N ∀i = 1, . . . , n}
n

# {(a, g) ∈ ΓE : kgk ≤ N } < ∞.

Applying (10.3.6) completes the proof. If G is only weakly separated, then for
all ρ > 0 the Schottky product hGa id(o,Ua )≤ρ is still stronglly separated, which is
enough to apply (10.3.6) in this context.



Proposition 10.3.7.
(i) If #(E) < ∞ and the groups Ga satisfy δGa < ∞, then δG < ∞.

(ii) Suppose that for some a ∈ E, Ga is of divergence type. Then δG > δGa .
(iii) Suppose that G is a group. If δGa = ∞ for some a, and if Gb is infinite
for some b 6= a, then δeG = ∞.
(iv) If E = {a, b} and Gb = g Z , then

lim δ(Ga ∗ g nZ ) = δ(Ga ).

n→∞

Moreover, if Ga is of convergence type, then for all n sufficiently large,
Ga ∗ g nZ is of convergence type.
Moreover, (ii) holds for any free product G = hGa ia∈E , even if the product is not
Schottky.

Remark 10.3.8. Property (iii) tells us that an analogue of property (i) cannot
hold for the modified Poincaré exponent: if we take the Schottky product of two
e i ) < ∞ but δ(Gi ) = ∞, then the product G will have
groups G1 , G2 with δ(G
e
δ(G) = ∞.
Proof of (i). (10.3.9) shows that for some C > 0,
#{g ∈ G : kgk ≤ ρ} ≤ # {(a, g) ∈ ΓE : kgk ≤ Cρ}
Applying (8.1.2) completes the proof.

Cρ

∀ρ > 0.


10.3. STRONGLY SEPARATED SCHOTTKY PRODUCTS

165

Proof of (ii). For all s ≥ 0,
X
X
P|g|
Σs (G) =
b−skπ(g)k ≥
b−s 1 kgi k
g∈Γ

g∈Γ

=

|g|
XY

b−skgi k

g∈Γ i=1

=

∞
X

X

X

n=0 a1 6=···6=an g1 ∈Ga1 \{id}

=

∞
X

X

n
Y

···

X

X

n
Y

b−skgi k

gn ∈Gan \{id} i=1

b−skgk

n=0 a1 6=···6=an i=1 g∈Gai \{id}

=

∞
X

X

n
Y

(Σs (Gai ) − 1).

n=0 a1 6=···6=an i=1

To simplify further calculations, we will assume that #(E) = 2; specifically we will
let E = {1, 2}. Then

!n/2

Y



2
Σs (Ga ) − 1
n even
∞ 

X
a∈E
Σs (G) ≥
!(n−1)/2
!
 Y
X


n=0 


Σs (Ga ) − 1
n odd
Σs (Ga ) − 1


a∈E

≍×

∞
X

n=0

Y

a∈E

Σs (Ga ) − 1

!n/2


a∈E

.

This series diverges if and only if
Y
(10.3.10)
(Σs (Ga ) − 1) ≥ 1.
a∈E

Now suppose that G1 is of divergence type, and let δ1 = δ(G1 ). By the monotone
convergence theorem,
Y
Y
lim
(Σs (Ga ) − 1) =
(Σδ1 (Ga ) − 1) = ∞(Σδ1 (G2 ) − 1) = ∞.
sցδ1

a∈E

a∈E

(The last equality holds since G2 is nontrivial, see Definition 10.2.1.) So for s
sufficiently close to δ1 , (10.3.10) holds, and thus Σs (G) = ∞.

Proof of (iii). Fix ρ > 0, and let h ∈ Gb satisfy d(h(o), Ua ) ≥ ρ. (This is
possible since Gb is non-elliptic and d(h(o), Ua ) ≍+ khk ∀h ∈ Gb .) Then the set
Sρ = {gh(o) : g ∈ Ga }
is ρ-separated, but δ(Sρ ) = δ(Ga ) = ∞. Since ρ was arbitrary, it follows from
e
Proposition 8.2.4(iv) that δ(G)
= ∞.


166

10. SCHOTTKY PRODUCTS

Proof of (iv). We will in fact show the following more general result:
Proposition 10.3.9. Suppose E = {a, b}, and fix s ∈
/ ∆(Ga ) ∪ ∆(Gb ). Then

there exists a finite set F ⊆ Gb such that for all H ≤ Gb , if H ∩ F = {id}, then
s∈
/ ∆(Ga ∗ H).
Indeed, for such an s, the Poincaré series Σs (Ga ∗H) can be estimated using (10.3.5)

as follows:

Σs (Ga ∗ H) =

X

g∈Γ

b−skπ(g)k ≤

X

b−s

P|g|
1

(kgi k−C)

g∈Γ

=

X

bsC|g|b−s

P|g|
1

kgi k

.

g∈Γ

Continuing as in the proof of part (ii), we get
Σs (Ga ∗ H) .×

∞
X

bsCn

n=0




n/2
Σs (Ga ) − 1 Σs (H) − 1
.

Since Σs (H) − 1 ≤ Σs (Gb \ F ), to show that Σs (Ga ∗ H) < ∞ it suffices to show

that

(10.3.11)

bsC




1/2
Σs (Ga ) − 1 Σs (Gb \ F )
< 1.

But since the series Σs (Ga ) and Σs (Gb ) both converge by assumption, (10.3.11)
holds for all F ⊆ Gb sufficiently large.



We will sometimes find the following variant of Proposition 10.3.7(ii) more
useful than the original:
Proposition 10.3.10 (Cf. [55, Proposition 2]). Fix H ≤ G ≤ Isom(X), and
suppose that
(I) ΛH $ ΛG ,
(II) G is of general type, and
(III) H is of compact type and of divergence type.
Then δG > δH .
Proof. Fix ξ ∈ ΛG \ ΛH , and fix ε > 0 small enough so that B(ξ, ε)∩ΛH = .
Since G is of general type, by Proposition 7.4.7, there exists a loxodromic isometry
g ∈ G such that g+ , g− ∈ B(ξ, ε/4). After replacing g by an appropriate power, we
may assume that g n (bord X \ B(ξ, ε/2)) ⊆ B(ξ, ε/2) for all n ∈ Z \ {0}. Now let
U1 = B(ξ, ε/2)
U2 = Nε/4 (ΛH ).
Since H is of compact type (and strongly discrete by Observation 8.1.5), Proposition
7.7.4 shows that there exists a finite set F ⊆ H such that for all h ∈ H \ F ,

10.3. STRONGLY SEPARATED SCHOTTKY PRODUCTS

h(bord X \ U2 ) ⊆ U2 . Let
S=

a

167

n
(H \ F ) × (g Z \ {id}) ,

n≥0

and define π : S → G via the formula


π (hi , ji )ni=1 = h1 j1 · · · hn jn .

A variant of the Ping-Pong Lemma shows that π is injective. On the other hand,
for all (hi , ji )ni=1 ∈ S, the triangle inequality gies

n
  X
[khi k + kji k].
d o, π (hi , ji )ni=1 (o) ≤
i=1

Thus for all s > δH ,

Σs (G) ≥
≥
=

X

g∈π(S)

X

(hi ,ji )n
i=1 ∈S

X

n≥0

=

e−skgk

X




n≥0


= ∞
< ∞

X

n
Y

e−s[khi k+kji k]

i=1

X

h∈H\F j∈gZ \{id}

n

e−s[khk+kjk] 

n
Σs (H \ F )Σs (g Z \ {id})

Σs (H \ F )Σs (g Z \ {id}) ≥ 1

Σs (H \ F )Σs (g Z \ {id}) < 1

.

Now since H is of divergence type, by the monotone convergence theorem,
lim Σs (H \ F )Σs (g Z \ {id}) = ΣδH (H \ F )ΣδH (g Z \ {id})

sցδH

= ∞ · (positive constant)
= ∞.
Thus, for s sufficiently close to δH , Σs (G) = ∞. This shows that δG > δH .



Remark 10.3.11. The reason that we couldn’t deduce Proposition 10.3.10 directly from Proposition 10.3.7(ii) is that the group hH, g Z i considered in the proof

of Proposition 10.3.10 is not necessarily a free product due to the existence of the
finite set F . In the Standard Case, this could be solved by taking a finite-index

subgroup of H whose intersection with F is trivial, but in general, it is not clear
that such a subgroup exists.

168

10. SCHOTTKY PRODUCTS

10.4. A partition-structure–like structure
For each g ∈ Γ, let

Wg = π(g)(Xa|g| ),

unless g = , in which case let W∅ = bord X.
Standing Assumption 10.4.1. In what follows, we assume that for each a ∈
E, either
(1) Ga is a group, or
(2) Ga ≡ N.
For g, h ∈ Γ, write g ≤ h if h = g ∗ k for some k ∈ Γ.
Lemma 10.4.2. Fix g, h ∈ Γ. If g ≤ h, then Wh ⊆ Wg . On the other hand, if

g and h are incomparable (g 6≤ h and h 6≤ g), then Wg ∩ Wh = .

Proof. The first assertion follows from Lemma 10.2.2. For the second assertion, it suffices to show that if (a, g), (b, h) ∈ ΓE are distinct, then W(a,g) ∩ W(b,h) =

. Since W(a,g) ⊆ Ua and (Ua )a∈E are disjoint, if a 6= b then W(a,g) ∩W(b,h) = . So

suppose a = b. Assumption 10.4.1 guarantees that either g −1 h ∈ Ga or h−1 g ∈ Ga ;
without loss of generality assume that g −1 h ∈ Ga . Then
W(a,g) ∩ W(b,h) = g(Xa ) ∩ h(Xa ) = g(Xa ∩ g −1 h(Xa )) ⊆ g(Xa ∩ Ua ) = .


Lemma 10.4.3. There exists σ > 0 such that for all g ∈ Γ,
(10.4.1)

Wg ⊆ Shad(π(g)(o), σ).

In particular
(10.4.2)

Diam(Wg ) .× b−kπ(g)k .

Proof. Let n = |g|, g = gn , a = an , and z = π(g)−1 (o). Observe that

if g(z) ∈ Va , then Lemma 10.2.2 implies that o ∈ Va1 , a contradiction. Thus
z ∈ g −1 (X\Va ). If X is not global, then (10.3.1)Ua =Va ,ε=ε/2 implies that D(z, Xa ) ≥
ε/2. On the other hand, if X is global then we have z ∈ Ua , so (10.3.2) implies

that D(z, Xa ) ≥ ε/2. Either way, we have D(z, Xa ) ≥ ε/2.
Let σ > 0 be large enough so that the Big Shadows Lemma 4.5.7 holds; then

we have Xa ⊆ Shadz (o, σ). Applying π(g) yields (10.4.1), and combining with the
Diameter of Shadows Lemma 4.5.8 yields (10.4.2).

Let ∂Γ denote the set of all infinite words with letters in ΓE such that ai 6= ai+1
for all i. Given g ∈ ∂Γ, for each n, g ↿ n ∈ Γ. Then Lemmas 10.4.2 and 10.4.3 show
that the sequence (Wg↿n )∞
0 is an infinite descending sequence of closed sets with

10.4. A PARTITION-STRUCTURE–LIKE STRUCTURE

diameters tending to zero; thus there exists a unique point ξ ∈
will be denoted π(g).

169

T∞
0

Wg↿n , which

Lemma 10.4.4. For all g ∈ ∂Γ, π(g ↿ n)(o) → π(g) radially. In particular

π(∂Γ) ⊆ Λr (G).

Proof. This is immediate from (10.4.1), since by definition π(g) ∈ Wg↿n for
all n.

Lemma 10.4.5 (Cf. Klein’s combination theorem [121, Theorem 1.1], [115]).
The set
(10.4.3)

D = bord X \

[ [

a∈E g∈Ga

g(Xa ) = bord X \

[

W(a,g)

(a,g)∈ΓE

satisfies G(D) = bord X \ π(∂Γ).
Before we begin the proof of this lemma, note that since D ∩ X is open (Lemma
10.4.6 below), the connectedness of X implies that g(D) ∩ D =
6  for some g ∈ G.
Thus D is not a fundamental domain.

Proof. Fix x ∈ bord X \ π(∂Γ), and consider the set
Γx := {g ∈ Γ : x ∈ Wg } .
By Lemma 10.4.2, Γx is totally ordered as a subset of Γ. If Γx is infinite, let g ∈ ∂Γ

be the unique word such that Γx = {g ↿ n : n ∈ N ∪ {0}}; Lemma 10.4.3 implies
that x = π(g) ∈ π(∂Γ), contradicting our hypothesis. Thus Γx is finite. If Γx = ,

we are done. Otherwise, let g be the largest element of Γx . Then x ∈ Wg , so
π(g)−1 (x) ∈ Xa , where a = a|g| . The maximality of g implies that
π(g)−1 (x) ∈
/ W(b,h) = h(Xb ) ∀b ∈ E \ {a} ∀h ∈ Gb \ {id},
but on the other hand π(g)−1 (x) ∈ Xa ⊆ bord X \ Ua implies that π(g)−1 (x) ∈
/
W(a,h) for all h ∈ Ga \ {id}. Thus π(g)−1 (x) ∈ D.



Lemma 10.4.6. Suppose that for each a ∈ E, Ga is strongly discrete. Then
[
D \ Int(D) ⊆
Λa ,
a∈E

where Λa = Λ(Ga ). In particular, D ∩ X is open.
Proof. Fix x ∈ D \ Int(D), and find a sequence (an , gn ) ∈ ΓE such that

D(x, gn (Xan )) → 0. Since gn (Xan ) ⊆ Uan , (10.3.1) implies that an is constant for

all sufficiently large n, say an = a. On the other hand, if there is some g ∈ Ga
such that gn = g for infinitely many n, then since g(Xa ) is closed we would have
x ∈ g(Xa ), contradicting that x ∈ D. Since Ga is strongly discrete, it follows that

170

10. SCHOTTKY PRODUCTS

kgn k → ∞, and thus Diam(gn (Xa )) → 0 by Lemma 10.4.3. Since gn (o) ∈ gn (Xa ),
it follows that gn (o) → x, and thus x ∈ Λa .
Theorem 10.4.7.
Λ = π(∂Γ) ∪

[ [



g(Λa ).

g∈G a∈E

Proof. The ⊇ direction follows from Lemma 10.4.4, so let us show ⊆. It
S
suffices to show that Λ ∩ D ⊆ a∈E Λa . Indeed, for all g ∈ Γ \ {}, Lemma 10.2.2
S
gives π(g)(o) ∈ g1 (Xa1 ) ⊆ bord X \D. Thus Λ∩D = D ∩∂D ⊆ a∈E Λa by Lemma

10.4.6.



Corollary 10.4.8. If E is finite and each Ga is strongly discrete and of compact type, then G is strongly discrete and of compact type.
Proof. Strong discreteness follows from Corollary 10.3.6, so let us show that
G is of compact type. Let (ξn )∞
1 be a sequence in Λ. For each n ∈ N, if ξn ∈ π(∂Γ),
write ξn = π(gn ) for some gn ∈ ∂Γ; otherwise, write ξn = π(gn )(ηn ) for some
gn ∈ Γ and ηn ∈ Λan . Either way, note that ξn ∈ Wh for all h ≤ g.
For each h ∈ Γ, let
Sh = {n ∈ N : h ≤ gn }.

Since Γ is countable, by extracting a subsequence we may without loss of generality
assume that for all h ∈ Γ, either n ∈ Sh for all but finitely many n, or n ∈ Sh for
only finitely many n. Let
Γ′ = {h ∈ Γ : n ∈ Sh for all but finitely many n}.
Then by Lemma 10.4.2, the set Γ′ is totally ordered. Moreover,  ∈ Γ′ . If Γ′ is
infinite, then choose g ∈ ∂Γ such that Γ′ = {g ↿ m : m ≥ 0}; by Lemma 10.4.3,

we have ξn → π(g). Otherwise, let g be the largest element of Γ′ . For each n,
either ξn ∈ Wg(bn ,hn ) for some (bn , hn ) ∈ ΓE , or ξn = π(g)(ηn ) for some an ∈ E

and ηn ∈ Λan . By extracting another subsequence, we may assume that either the
first case holds for all n, or the second case holds for all n. Suppose the first case
holds for all n. The maximality of g implies that for each (b, h) ∈ ΓE , there are
only finitely many n such that (bn , hn ) = (b, h). Since E is finite, by extracting

a further subsequence we may assume that bn = b for all n. Since Gb is strongly
discrete and of compact type, by extracting a further subsequence we may assume
that hn (o) → η for some η ∈ Λb . But then ξn → π(g)(η) ∈ Λ.
S
Suppose the second case holds for all n. Since ΛE = a∈E Λa is compact, by

extracting a further subsequence we may assume that ηn → η for some η ∈ ΛE .
But then ξn → π(g)(η) ∈ Λ.

Corollary 10.4.9. If #(ΓE ) ≥ 3, then #(Λ) ≥ #(R).

10.5. EXISTENCE OF SCHOTTKY PRODUCTS

171

Proof. The hypothesis #(ΓE ) ≥ 3 implies that for each g ∈ Γ,
X
#{(a, g)(b, h) ∈ Γ2E : g(a, g)(b, h) ∈ Γ} =
(#(Ga ) − 1)(#(Gb ) − 1) ≥ 2.
b6=a6=a|g|

Thus, the tree Γ has no terminal nodes or infinite isolated paths. It follows that
∂Γ is perfect and therefore has cardinality at least #(R); since π(∂Γ) ⊆ Λ, we have
#(Λ) ≥ #(∂Γ) ≥ #(R).

Proposition 10.4.10. Suppose that the Schottky system (Ua )a∈E is global.
Then if (Ga )a∈E are moderately (resp. weakly) discrete, then G is moderately (resp.
weakly) discrete. If (Ga )a∈E act properly discontinuously, then G acts properly discontinuously.
Proof. Let D be as in (10.4.3). Fix x ∈ D and g ∈ Γ, let n = |g|, and suppose
that π(g)(x) ∈ D. Then:
(A) For all i = 1, . . . , n, if gi+1 · · · gn (x) ∈ Xai then Lemma 10.2.2 would give
π(g)(x) ∈ g1 (Xa1 ), so gi+1 · · · gn (x) ∈ Vai .

(B) For all i = 0, . . . , n − 1, if gi+1 · · · gn (x) ∈ Xai+1 then Lemma 10.2.2 would
give x ∈ gn−1 (Xan ), so gi+1 · · · gn (x) ∈ Vai+1 .

If n ≥ 2, letting i = 1 in (A)-(B) yields a contradiction, so n = 0 or 1. Moreover,
if n = 1, plugging in i = 1 in (A) gives x ∈ Va1 .
To summarize, if we let

Gx =
then


G

a

{id}

x ∈ Va
S
x∈
/ a∈E Va

g(x) ∈ D ⇒ g ∈ Gx ∀g ∈ G.
More concretely,
d(x, g(x)) < d(x, X \ D) ⇒ g ∈ Gx ∀g ∈ G.
Comparing with the definitions of moderate and weak discreteness (Definition 5.2.1)
and the definition of proper discontinuity (Definition 5.2.11) completes the proof.

10.5. Existence of Schottky products
Proposition 10.5.1. Suppose that ΛIsom(X) = ∂X, and let G1 , G2 ≤ Isom(X)

be groups with the following property: For i = 1, 2, there exist ξi ∈ ∂X and ε > 0

such that
(10.5.1)

D(ξi , g(ξi )) ≥ ε ∀g ∈ Gi \ {id}.

172

10. SCHOTTKY PRODUCTS

Then there exists φ ∈ Isom(X) such that the product hG1 , φ(G2 )i is a global strongly

separated Schottky product.

Proof. We begin the proof with the following
Claim 10.5.2. For each i = 1, 2, there exists an open set Ai ∋ ξi such that
g(Ai ) ∩ Ai =  for all g ∈ Gi \ {id}.
Proof. Fix i = 1, 2. Clearly, (10.5.1) implies that ξi ∈
/ Λ(Gi ). Thus, there
exists δ > 0 such that D(g(o), ξi ) ≥ δ for all g ∈ Gi . Applying the Big Shadows

Lemma 4.5.7, there exists σ > 0 such that B(ξi , δ/2) ⊆ Shadg−1 (o) (o, σ) for all
g ∈ G. But then by the Bounded Distortion Lemma 4.5.6, we have
Diam(g(B(ξi , γ))) ≍×,σ b−kgk Diam(B(ξi , γ)) ≤ 2γ ∀g ∈ G ∀0 < γ ≤ δ/2.
Choosing γ appropriately gives Diam(g(B(ξi , γ))) < ε/2 for all g ∈ G. Letting
Ai = B(ξi , γ) completes the proof of the claim.

⊳

For each i = 1, 2, let Ai be as above, and fix an open set Bi ∋ ξi such that
D(Bi , bord X \ Ai ) > 0. Since ΛIsom(X) = ∂X, there exists a loxodromic isometry

φ ∈ Isom(X) such that φ− ∈ B1 and φ+ ∈ B2 (Proposition 7.4.7). Then by
Theorem 6.1.10, φn → φ+ uniformly on bord X \ B1 , so φn (B1 ) ∪ B2 = bord X
for all n ∈ N sufficiently large. Fix such an n, and let U1 = φn (bord X \ A1 ),
U2 = bord X \ A2 , V1 = φn (bord X \ B1 ), V2 = bord X \ B2 . Then (V1 , V2 ) is

a global Schottky system for (G1 , φ(G2 )), which implies that (U1 , U2 ) is a global
strongly separated Schottky system for (G1 , φ(G2 )). This concludes the proof of
the proposition.

Remark 10.5.3. The hypotheses of the above theorem are satisfied if X is
an algebraic hyperbolic space and for each i = 1, 2, Gi is strongly discrete and of
compact type and Λi = Λ(Gi ) $ ∂X.
Proof. We have ΛIsom(X) = ∂X by Observation 2.3.2. Fix i = 1, 2. Since
Λi $ ∂X, ∂X \ Λ(Gi ) is a nonempty open set. For each g ∈ Gi \ {id}, the set
Fix(g) is totally geodesic (Theorem 2.4.7) and therefore nowhere dense; since Gi is
S
countable, it follows that g∈Gi \{id} Fix(g) is a meager set, so the set
[

Fix(g)
Si = ∂X \ Λ(Gi ) \
g∈Gi \{id}

is nonempty. Fix ξi ∈ Si . By Proposition 7.7.4,
lim inf D(ξi , g(ξi )) ≥ D(ξi , Λ(Gi )) > 0.
g∈Gi

10.5. EXISTENCE OF SCHOTTKY PRODUCTS

173

On the other hand, for all g ∈ Gi \ {id} we have ξi ∈
/ Fix(g) and therefore
D(ξi , g(ξi )) > 0. Combining yields inf g∈Gi \{id} D(ξi , g(ξi )) > 0 and thus (10.5.1).
This concludes the proof of the remark.

Proposition 10.5.4. For a semigroup G  Isom(X), the following are equiv-

alent:

(A) G is either outward focal or of general type.
(B) G contains a strongly separated Schottky subsemigroup.
e
(C) δ(G)
> 0.

(D) #(ΛG ) ≥ #(R).

(E) #(ΛG ) ≥ 3, i.e. G is nonelementary.

If G is a group, then these are also equivalent to:
(F) G contains a global strongly separated Schottky subgroup.
The implications (C) ⇒ (E) ⇒ (A) have been proven elsewhere in the paper; see

Proposition 7.3.1 and Theorem 1.2.3. The implication (B) ⇒ (D) is an immediate
consequence of Corollary 10.4.9, and (D) ⇒ (E) and (F) ⇒ (B) are both trivial.

So it remains to prove (A) ⇒ (B) ⇒ (C), and that (A) ⇒ (F) if G is a group.

Proof of (A) ⇒ (B). Suppose first that G is outward focal with global fixed

point ξ. Then there exists g ∈ G with g ′ (ξ) > 1, and there exists h ∈ G such that
h+ 6= g+ . If we let j = g n h, then j ′ (ξ) > 1 (after choosing n sufficiently large), and

j+ 6= g+ .

So regardless of whether G is outward focal or of general type, there exist
loxodromic isometries g, h ∈ G such that g+ ∈
/ Fix(h) and h+ ∈
/ Fix(g). It follows
that there exists ε > 0 such that


B(g+ , ε) ∩ hn B(g+ , ε) = B(h+ , ε) ∩ g n B(h+ , ε) =  ∀n ≥ 1.

Let U1 = B(g+ , ε/2), U2 = B(h+ , ε/2), V1 = B(g+ , ε), and V2 = B(h+ , ε). By
Theorem 6.1.10, for all sufficiently large n we have g n (V1 ∪ V2 ) ⊆ U1 and hn (V1 ∪
V2 ) ⊆ U2 . It follows that (V1 , V2 ) is a Schottky system for ((g n )N , (hn )N ), and that
(U1 , U2 ) is a strongly separated Schottky system for ((g n )N , (hn )N ).


Proof of (B) ⇒ (C). Since a cyclic loxodromic semigroup is of divergence
type (an immediate consequence of (6.1.3)), Proposition 10.3.7(i),(ii) shows that
e
0 < δ(H) < ∞, where H  G is a Schottky subsemigroup. Thus δ(H)
> 0, and so
e
δ(G)
> 0.



Proof of (A) ⇒ (F) for groups. Fix loxodromic isometries g, h ∈ G with

Fix(g) ∩ Fix(h) = . Choose ε > 0 such that


B(Fix(g), ε) ∩ hn B(Fix(g), ε) = B(Fix(h), ε) ∩ g n B(Fix(h), ε) =  ∀n ≥ 1.

174

10. SCHOTTKY PRODUCTS

Let U1 = B(Fix(g), ε/2), U2 = B(Fix(h), ε/2), V1 = B(Fix(g), ε), and V2 =
B(Fix(h), ε). By Theorem 6.1.10, for all sufficiently large n we have g n (bord X \
B(g− , ε/2)) ⊆ B(g+ , ε/2) and hn (bord X \ B(h− , ε/2)) ⊆ B(h+ , ε/2). It follows
that (V1 , V2 ) is a global Schottky system for ((g n )Z , (hn )Z ), and that (U1 , U2 ) is a
global strongly separated Schottky system for ((g n )Z , (hn )Z ).



CHAPTER 11

Parabolic groups
In this chapter we study parabolic groups. We begin with a list of several examples of parabolic groups acting on E∞ , the half-space model of infinite-dimensional
real hyperbolic geometry. These examples include a counterexample to the infinitedimensional analogue of Margulis’s lemma, as well as a parabolic isometry that
generates a cyclic group which is not discrete in the ambient isometry group. The
latter example is the Poincaré extension of an example due to M. Edelstein. After
giving these examples of parabolic groups, we prove a lower bound on the Poincaré
exponent of a parabolic group in terms of its algebraic structure (Theorem 11.2.6).
We show that it is optimal by constructing explicit examples of parabolic groups
acting on E∞ which come arbitrarily close to this bound.
11.1. Examples of parabolic groups acting on E∞
Let X = E = E∞ be the half-space model of infinite-dimensional real hyperbolic
geometry (§2.5.2). Recall that B = ∂E \ {∞} is an infinite-dimensional Hilbert
space, and that Poincaré extension is the homomorphism b· : Isom(B) → Isom(E)
given by the formula

b· (g)(t, x) = b
g(t, x) = (t, g(x))

(Observation 2.5.6). The image of b· is the set {g ∈ Stab(Isom(E); ∞) : g ′ (∞) = 1}.

Thus, Poincaré extension provides a bijection between the class of subgroups of
Isom(B) and the class of subgroups of Isom(E) for which ∞ is a neutral global fixed
b We may
point. Given a group G ≤ Isom(B), we will denote its image under b· by G.
b and G as follows:
summarize the relation between G
Observation 11.1.1.

b is parabolic if and only if G(0) is unbounded; otherwise G
b is elliptic.
(i) G
b is strongly (resp. moderately, weakly, COT) discrete if and only if G
(ii) G
b acts properly discontinuously if and only if G does.
is. G

(iii) Write Isom(B) = O(B) ⋉ B. Then the preimage of the uniform operator
topology under b· is equal to the product of the uniform operator topology

on O(B) with the usual topology on B. Thus if we denote this topology
b is UOT-discrete if and only if G is UOT*-discrete.
by UOT*, then G
175

176

11. PARABOLIC GROUPS

(iv) For all g ∈ G, we have
ekbgk ≍× cosh kb
gk = 1 +

k(1, g(0)) − (1, 0)k2
≍× 1 ∨ kg(0)k2
2

and thus for all s ≥ 0,
(11.1.1)

b ≍× Σs (G) :=
Σs (G)

X

g∈G

(1 ∨ kg(0)k)−2s .

b and we say that G
In what follows, we let δ(G) = inf{s : Σs (G) < ∞} = δ(G),
b
is of convergence or divergence type if G is.

11.1.1. The Haagerup property and the absence of a Margulis lemma.
One question which has been well studied in the literature is the following: For
which abstract groups Γ can Γ be embedded as a strongly discrete subgroup of
Isom(B)? Such a group is said to have the Haagerup property.1 For a detailed
account, see [50].
Remark 11.1.2. The following groups have the Haagerup property:
• [62, pp.73-74] Groups which admit a cocompact action on a proper R-tree.

In particular this includes Fn (Z) for every n.
• [101] Amenable groups. This includes solvable and nilpotent groups.

A class of examples of groups without the Haagerup property is the class of infinite
groups with Kazdan’s property (T). For example, if n ≥ 3 then SLn (Z) does not
have the Haagerup property [22, §4.2].
The example of (virtually) nilpotent groups will be considered in more detail
in §11.2.3, since it turns out that every parabolic subgroup of Isom(E) which has
finite Poincaré exponent is virtually nilpotent.
Recall that Margulis’s lemma is the following lemma:

Proposition 11.1.3 (Margulis’s lemma, [61, p.126] or [15, p.101]). Let X be
a Hadamard manifold with curvature bounded away from −∞. Then there exists

ε = εX > 0 with the following property: For every discrete group G ≤ Isom(X) and
for every x ∈ X, the group
Gε (x) := hg ∈ G : d(x, g(x)) ≤ εi
is virtually nilpotent.

For convenience, we will say that Margulis’s lemma holds on a metric space
X if the conclusion of Proposition 11.1.3 holds, i.e. if there exists ε > 0 such
1The Haagerup property can also be defined for locally compact groups, by replacing “finite” with

“precompact” in the definition of strong discreteness. However, in this monograph we consider
only discrete groups.

11.1. EXAMPLES OF PARABOLIC GROUPS ACTING ON E∞

177

that for every strongly discrete group G ≤ Isom(X) and for every x ∈ X, Gε (x)

is virtually nilpotent. It was proven recently by E. Breuillard, B. Green, and T.
C. Tao [38, Corollary 1.15] that Margulis’s lemma holds on all metric spaces with
bounded packing in the sense of [38]. This result includes Proposition 11.1.3 as a
special case.
By contrast, in infinite dimensions we have the following:
Observation 11.1.4. Margulis’s lemma does not hold on the space X = E =
∞

E .
Proof. Since F2 (Z) has the Haagerup property, there exists a strongly discrete
b be the
group G ≤ Isom(B) isomorphic to F2 (Z), say G = (g1 )Z ∗ (g2 )Z . Let G
Poincaré extension of G. Fix ε > 0, and let
x = (t, 0) ∈ E
for t > 0 large to be determined. Then by (2.5.3),

d(x, gbi (x)) = d (t, 0), (t, gi (0)) ≍× kgi (0)k/t.

bε (x), and so
So if t is large enough, then d(x, gbi (x)) ≤ ε. It follows that b
g1 , b
g2 ∈ G
b ε (x) = G
b ≡ F2 (Z) is not virtually nilpotent.
G


Remark 11.1.5. In view of the fact that in the finite-dimensional Margulis’s

lemma, εEd depends on the dimension d and tends to zero as d → ∞ (see e.g.
[23, Proposition 5.2]), we should not be surprised that the lemma fails in infinite
dimensions.
Remark 11.1.6. In some references (e.g. [148, Theorem 12.6.1]), the conclusion of Margulis’s lemma states that Gε (x) is elementary rather than virtually
nilpotent. The above example shows that the two statements should not be confused with each other. We will show (Example 13.1.5 below) that the alternative
formulation of Margulis’s lemma which states that Gε (x) is elementary also fails in
infinite dimensions.
Remark 11.1.7. Parabolic groups acting on proper CAT(-1) spaces must be
amenable [41, Proposition 1.6]. Therefore the existence of a parabolic subgroup
of Isom(H∞ ) isomorphic to F2 (Z) also distinguishes H∞ from the class of proper
CAT(-1) spaces.
11.1.2. Edelstein examples. One of the oldest results in the field of groups
acting by isometries on Hilbert space is the following example due to M. Edelstein:

178

11. PARABOLIC GROUPS

Proposition 11.1.8 ([70, Theorem 2.1]). There exist an isometry g belonging
(1)

(2)

∞
to Isom(ℓ2 (N; C)) and sequences (nk )∞
1 , (nk )1 such that
(1)

(2)

g nk (0) −
→ 0 but kg nk (0)k −
→ ∞.

(11.1.2)

k

k

Since the specific form of Edelstein’s example will be important to us, we recall
the proof of Proposition 11.1.8, in a modified form suitable for generalization:

Proof of Proposition 11.1.8. For each k ∈ N let ak = 1/k!, let bk = 1,

and let

ck = e2πiak , dk = bk (1 − ck ).

(11.1.3)
Then

X

k∈N

X

|dk |2 .×

k∈N

|ak bk |2 =

X  1 2

k∈N

k!

< ∞,

2
2
so d = (dk )∞
1 ∈ ℓ (N; C). Let g ∈ Isom(ℓ (N; C)) be given by the formula

(11.1.4)

g(x)k = ck xk + dk .

Then
(11.1.5)

g n (x)k = cnk xk +

n−1
X

cik dk = cnk xk +

i=0

n

1 − cnk
dk = cnk xk + bk (1 − cnk ).
1 − ck

In particular, g (0)k = bk (1 − cnk ). So
(11.1.6)
∞
∞
∞
X
X
X
n
2
n 2
2πinak 2
kg (0)k =
|bk (1 − ck )| =
|bk (1 − e
)| ≍×
|bk |2 d(nak , Z)2 .
k=1

k=1

k=1

Now for each k ∈ N, let

(1)

nk = k!
k

(2)

nk =
Then
kg

(1)

nk

1X
j!
2 j=1

2

∞  2
∞
X
X
1
k!
2
, Z = (k!)
d
(0)k ≍×
j!
j!
j=1
j=k+1

2
k!
≍×
(k + 1)!
1
−
→ 0,
=
(k + 1)2 k
2

11.1. EXAMPLES OF PARABOLIC GROUPS ACTING ON E∞

179

but on the other hand
kg

(2)

nk

2

(0)k &×

k
X
j=1

d

"
#2
!2
j
k
(2)
X
X
nk
i!
1
1−
,Z
=
(j + 1)!
4
(j + 1)!
i=1
j=1
≥

2
k 
2
1X
1−
−
→ ∞.
k
4 j=1
j+1

This demonstrates (11.1.2).



Remark 11.1.9. Let us explain the significance of Edelstein’s example from
the point of view of hyperbolic geometry. Let g ∈ Isom(B) be as in Proposition
11.1.8, and let gb ∈ Isom(E∞ ) be its Poincaré extension. By Observation 11.1.1, b
g is

a parabolic isometry. But the orbit of o = (1, 0) (cf. §4.1) is quite irregular; indeed,
(1)

(2)

gb nk (o) −
→ o but gb nk (o) −
→ ∞ ∈ ∂E.
k

n

(o))∞
1

k

So the orbit (g
simultaneously tends to infinity on one subsequence while
remaining bounded on another subsequence. Such a phenomenon cannot occur in
proper metric spaces, as we demonstrate now:
Theorem 11.1.10. If X is a proper metric space and if G ≤ Isom(X) is cyclic,

then either G has bounded orbits or G is strongly discrete.

Proof. Write G = g Z for some g ∈ Isom(X), and fix a point o ∈ X. For each
n ∈ Z write knk = kg n k. Then k − nk = knk, and km + nk ≤ kmk + knk.
Suppose that G is not strongly discrete. Then there exists R > 0 such that

(11.1.7)

#{n ∈ N : knk ≤ R} = ∞.

Now let g S (o) ⊆ g Z (o) ∩ B(o, 2R) be a maximal R-separated set. Since X is proper,
S is finite. For each k ∈ S, choose ℓk > k such that kℓk k ≤ R; such an ℓk exists by

(11.1.7).
Now let n ∈ N be arbitrary. Let 0 ≤ m ≤ n be the largest number for which
kmk ≤ 2R. Since g S (o) is a maximal R-separated subset of g Z (o) ∩ B(o, 2R) ∋
g m (o), there exists k ∈ S for which km − kk ≤ R. Then
km − k + ℓk k ≤ R + R = 2R.
On the other hand, m − k + ℓk > m since ℓk > k by construction. Thus by the

maximality of m, we have m − k + ℓk > n. So

n − m < ℓk − k ≤ C := max(ℓk − k).
k∈S

It follows that
knk ≤ kmk + kn − mk ≤ 2R + Ckgk,
i.e. knk is bounded independent of n. Thus G has bounded orbits.



180

11. PARABOLIC GROUPS

At this point, we shall use the different notions of discreteness introduced in
Chapter 5 to distinguish between different variations of Edelstein’s example. To
this end, we make the following definition:
Definition 11.1.11. An isometry g ∈ Isom(ℓ2 (N; C)) is said to be Edelstein∞
type if it is of the form (11.1.4), where the sequences (ck )∞
1 and (dk )1 are of
∞
the form (11.1.3), where (ak )∞
1 and (bk )1 are sequences of positive real numbers
satisfying
∞
X
|ak bk |2 < ∞.
k=1

Our proof of Proposition 11.1.8 shows that the isometry g is always well-defined
and satisfies (11.1.6). On the other hand, the conclusion of Proposition 11.1.8 does
not hold for all Edelstein-type isometries; it is possible that the cyclic group G = g Z
is strongly discrete, and it is also possible that this group has bounded orbits. (But
the two cannot happen simultaneously unless g is a torsion element.) In the sequel,
we will be interested in Edelstein-type isometries for which G has unbounded orbits
but is not necessarily strongly discrete. We will be able to distinguish between the
examples using our more refined notions of discreteness.
Edelstein-type Example 11.1.12. In our notation, Edelstein’s original example can be described as the Edelstein-type isometry g defined by the sequences
ak = 1/k!, bk = 1. Edelstein’s proof shows that G = g Z has unbounded orbits and
is not weakly discrete. However, we can show more:
Proposition 11.1.13. The cyclic group generated by the isometry in Edelstein’s
example is not UOT-discrete.
Proof. As in the proof of Proposition 11.1.8, we let nk = k!, so that g nk (0) →
0. But if T n denotes the linear part of g n , then
T nk (x) = (e2πik!/j! xj )∞
j=1
and so
kT nk − Ik ≤

∞
X

j=k+1

|1 − e2πik!/j! | ≍×

1
→ 0.
k+1

Thus T nk → I in the uniform operator topology, so by Observation 11.1.1(iii),
gbnk → id in the uniform operator topology. Thus gbZ is not UOT-discrete.



Edelstein-type Example 11.1.14. The Edelstein-type isometry g defined

by the sequences ak = 1/2k , bk = 1. This example was considered by A. Valette
[174, Proposition 1.7]. It has unbounded orbits, and is moderately discrete (in fact
properly discontinuous) but not strongly discrete.

11.1. EXAMPLES OF PARABOLIC GROUPS ACTING ON E∞

181

(1)

Proof. Letting nk = 2k , we have by (11.1.6)
 k 2
∞
∞
X
X
(1)
2
1
d
kg nk (0)k2 ≍×
,
Z
=
(2k−j )2 =
j
2
3
j=1
j=k+1

(2)

and so g Z is not strongly discrete. Letting nk = ⌊2k /3⌋, we have
2
 k
2 X

k   k−j
∞
X
(2)
2
1
⌊2 /3⌋
d
d
,
Z
≥
,
Z
−
kg nk (0)k2 ≍×
2j
3
2j
j=1
j=1
≥

k
X
1
100
j=7

≍+,× k −
→∞
k

and therefore g Z has unbounded orbits.
Finally, we show that g Z acts properly discontinuously. To begin with, we
observe that for all n ∈ N, we may write n = 2k (2j + 1) for some j, k ≥ 0; then
2
2
 k
 k
∞
X
2 (2j + 1)
2 (2j + 1)
n
2
,Z ≥ d
, Z = 1/4,
d
kg (0)k ≍×
2i
2k+1
i=1
i.e. 0 is an isolated point of g Z (0). So for some ε > 0,
kg n (0)k ≥ ε ∀n ∈ N.
Now let x ∈ ℓ2 (N; C) be arbitrary, and let N be large enough so that
k(xN +1 , . . .)k ≤ ε/3.
Now for all n ∈ N, we have
N

kg 2

n

(x) − xk = kg 2

N

n

N

(0, . . . , 0, xN +1 , . . .) − (0, . . . , 0, xN +1 , . . .)k

≥ kg 2

n

(0)k − 2ε/3 ≥ ε/3

which implies that the set g 2N Z (x) is discrete. But g Z (x) is the union of finitely
N

many isometric images of g 2

Z

(x), so it must also be discrete.



Remark 11.1.15. It is not possible to differentiate further between unbounded
Edelstein-type isometries by considering separately the conditions of weak discreteness, moderate discreteness, and proper discontinuity. Indeed, if X is any metric
space and if G ≤ Isom(X) is any cyclic group with unbounded orbits, then the
following are equivalent: G is weakly discrete, G is moderately discrete, G acts
properly discontinuously. This can be seen as follows: every nontrivial subgroup
of G is of finite index, and therefore also has unbounded orbits; it follows that no
element of G\ {id} has a fixed point in X; it follows from this that the three notions
of discreteness are equivalent.

182

11. PARABOLIC GROUPS

Example 11.1.16. Let g ∈ Isom(ℓ2 (N; C)) be as in Proposition 11.1.8, let σ :

ℓ2 (Z; C) → ℓ2 (Z; C) be the shift map σ(x)k = xk+1 , and let T : ℓ2 (N; C) → ℓ2 (N; C)
be given by the formula
T (x)k = e2πi/k xk .
Then g1 = g ⊕σ has unbounded orbits and is COT-discrete but not weakly discrete;
g2 = g ⊕ T has unbounded orbits and is UOT-discrete but not COT-discrete.

Proof. Since g has unbounded orbits and is not weakly discrete, the same
is true for both g1 and g2 . Since the sequence (σ n (x))∞
1 diverges for every x ∈

ℓ2 (Z; C), the group generated by σ is COT-discrete, which implies that g1 is as
well. Since the sequence (kT n − Ik)∞
1 is bounded from below, the group generated
by T is UOT-discrete, which implies that g2 is as well. On the other hand, if we
let nk = k!, then T nk (x) → x for all x ∈ ℓ2 (Z; C). But we showed in Proposition

11.1.13 that g nk (x) → x for all x ∈ ℓ2 (N; C); it follows that g ⊕ T is not COTdiscrete.

Remark 11.1.17. One might object to the above examples on the grounds that

the isometries g1 and g2 do not act irreducibly. However, Edelstein-type isometries
never act irreducibly: if g is defined by (11.1.3) and (11.1.4) for some sequences
∞
2
(ak )∞
1 and (bk )1 , then for every k the affine hyperplane Hk = {x ∈ ℓ (N; C) :
xk = bk } is invariant under g. In general, it is not even possible to find a minimal

subspace on which the restricted action of g is irreducible, since such a minimal
subspace would be given by the formula

P∞
2

\
1 |bk | = ∞
,
Hk =
P
∞
{(b )∞ }
|b |2 < ∞
k

k 1

1

k

and if g has unbounded orbits (as in Examples 11.1.12 and 11.1.14), the first case
must hold.
We conclude this section with one more Edelstein-type example:
Edelstein-type Example 11.1.18. The Edelstein-type isometry g defined by
the sequences ak = 1/2k , bk = log(1 + k). In this example, g Z is strongly discrete
but has infinite Poincaré exponent.
Proof. To show that g Z is strongly discrete, fix n ≥ 1, and let k be such that
2 ≤ n < 2k+1 . Then 1/4 ≤ n/2k+2 < 1/2, so by (11.1.6),
 n
2
bk+2
kg n (0)k2 &× bk+2 d k+2 , Z ≥
−
→ ∞.
2
16 n
k

11.2. THE POINCARÉ EXPONENT OF A FINITELY GENERATED PARABOLIC GROUP183

To show that δ(g Z ) = ∞, fix ℓ ≥ 0, and note that by (11.1.6),
ℓ

kg 2 (0)k2 ≍×

∞
X
4ℓ
|bk |2 ≍× |bℓ+1 |2 = log2 (2 + ℓ).
4k

k=ℓ+1

It follows that
Σs (g Z ) ≥

∞
∞
X
X
ℓ
(1 ∨ kg 2 (0)k)−2s ≍×
log−2s (2 + ℓ) = ∞ ∀s ≥ 0.
ℓ=0

ℓ=0



11.2. The Poincaré exponent of a finitely generated parabolic group
In this section, we relate the Poincaré exponent δG of a parabolic group G
with its algebraic structure. We will show below that δG is infinite unless G is
virtually nilpotent (Theorem 11.2.6 below), so we begin with a digression on the
coarse geometry of finitely generated virtually nilpotent groups.

11.2.1. Nilpotent and virtually nilpotent groups. Recall that the lower
central series of an abstract group Γ is the sequence (Γi )∞
1 defined recursively by
the equations
Γ1 = Γ and Γi+1 = [Γ, Γi ].
Here [A, B] denotes the commutator of two sets A, B ⊆ Γ, i.e. [A, B] = haba−1 b−1 :

a ∈ A, b ∈ Bi. The group Γ is nilpotent if its lower central series terminates, i.e. if
Γk+1 = {id} for some k ∈ N. The smallest integer k for which this equality holds
is called the nilpotency class of Γ, and we will denote it by k.

Note that a group is abelian if and only if it is nilpotent of class 1. The
fundamental theorem of finitely generated abelian groups says that if Γ is a finitely
generated abelian group, then Γ ≡ Zd × F for some d ∈ N ∪ {0} and some finite
abelian group F . The number d will be called the rank of Γ, denoted rank(Γ). Note
that the large-scale structure of Γ depends only on d and not on the finite group
F . Specifically, if dΓ is a Cayley metric on Γ then
(11.2.1)

NΓ (R) ≍× Rd ∀R ≥ 1.

Here NΓ (R) = #{γ ∈ Γ : d(e, γ) ≤ R} is the orbital counting function of Γ

interpreted as acting on the metric space (Γ, dΓ ) (cf. Remark 8.1.3).
The following analogue of (11.2.1) was proven by H. Bass and independently
by Y. Guivarch:

184

11. PARABOLIC GROUPS

Theorem 11.2.1 ([16, 87]). Let Γ be a finitely generated nilpotent group with
lower central series (Γi )∞
1 and nilpotency class k, and let
αΓ =

k
X

i rank(Γi /Γi+1 ).

i=1

Let dΓ be a Cayley metric on Γ. Then for all R ≥ 1,
(11.2.2)

NΓ (R) ≍× RαΓ .

The number αΓ will be called the (polynomial) growth rate of NΓ .

A group is virtually nilpotent if it has a nilpotent subgroup of finite index. The
following is an immediate corollary of Theorem 11.2.1:
Corollary 11.2.2. Let Γ be a finitely generated virtually nilpotent group. Let
′

Γ ≤ Γ be a nilpotent subgroup of finite index, and let dΓ be a Cayley metric. Let
αΓ = αΓ′ . Then for all R ≥ 1,
(11.2.3)

NΓ (R) ≍× RαΓ .

Example 11.2.3. If Γ is abelian, then (11.2.2) reduces to (11.2.1).
Example 11.2.4. Let Γ be the discrete Heisenberg group, i.e.





 1 a c



Γ= 
1 b  : a, b, c ∈ Z .




1

We compute the growth rate of NΓ . Note that Γ is nilpotent of class 2, and its
lower central series is given by Γ1 = Γ,


 1

Γ2 = 
1



c
1

Thus





:c∈Z .




αΓ = rank(Γ1 /Γ2 ) + 2 rank(Γ2 ) = 2 + 2 · 1 = 4.
Corollary 11.2.2 implies that finitely generated virtually nilpotent groups have
polynomial growth, meaning that the growth rate
(11.2.4)

αΓ := lim

R→∞

log NΓ (R)
log(R)

exists and is finite. The converse assertion is a deep theorem of M. Gromov:
Theorem 11.2.5 ([155]). A finitely generated group Γ has polynomial growth if
and only if Γ is virtually nilpotent. Moreover, if Γ does not have polynomial growth
then the limit (11.2.4) exists and equals ∞.

11.2. THE POINCARÉ EXPONENT OF A FINITELY GENERATED PARABOLIC GROUP185

Thus the limit (11.2.4) exists in all circumstances, so we may refer to it unambiguously.
11.2.2. A universal lower bound on the Poincaré exponent. Now let
G ≤ Isom(X) be a parabolic group. Recall that in the Standard Case, if a group
G is discrete then it is virtually abelian. Moreover, in this case δG = 12 rank(G).
If G is virtually nilpotent, then it is natural to replace this formula by the
formula δG = 21 αG . However, in general equality does not hold in this formula, as
we will see below (Theorem 11.2.11). We show now that the ≥ direction always
holds. Precisely:
Theorem 11.2.6. Let G ≤ Isom(X) be a finitely generated parabolic group. Let
αG be as in (11.2.4). Then
αG
(11.2.5)
δG ≥
·
2
Moreover, if equality holds and δG < ∞, then G is of divergence type.
Before proving Theorem 11.2.6, we make a few remarks:
Remark 11.2.7. In this theorem, it is crucial that b > 1 is chosen close enough
to 1 so that Proposition 3.6.8 holds (cf. §4.1). Indeed, by varying the parameter b

one may vary the Poincaré exponent at will (cf. (8.1.2)); in particular, by choosing
b large, one could make δG arbitrarily small. If X is strongly hyperbolic, then of

course we may let b = e.
Remark 11.2.8. Expanding on the above remark, we recall that if X is an
R-tree, then any value of b is permitted in Proposition 3.6.8 (Remark 3.6.12). This
demonstrates that if a finitely generated parabolic group acting on an R-tree has
finite Poincaré exponent, then its growth rate is zero. This may also be seen more
directly from Remark 6.2.12.
Remark 11.2.9. Let G ≤ Isom(X) be a group of general type, and let H ≤ G

be a finitely generated parabolic subgroup. Then combining Theorem 11.2.6 with
Proposition 10.3.10 shows that δG > αH /2. This generalizes a well-known theorem
of A. F. Beardon [18, Theorem 7].
Combining Theorems 11.2.5 and 11.2.6 gives the following corollary:
Corollary 11.2.10. Any finitely generated parabolic group with finite Poincaré
exponent is virtually nilpotent.
This corollary can be viewed very loosely as a generalization of Margulis’s lemma
(Proposition 11.1.3). As we have seen above (Observation 11.1.4), a strict analogue
of Margulis’s lemma fails in infinite dimensions.

186

11. PARABOLIC GROUPS

Proof of Theorem 11.2.6. Let g1 , . . . , gn be a set of generators for G, and
let dG denote the corresponding Cayley metric. Let ξ ∈ ∂X denote the unique fixed
point of G. Fix g ∈ G, and write g = gi1 · · · gim . By the universal property of path

metrics (Remark 3.1.4), we have

Dξ (o, g(o)) .× dG (id, g).
Now we apply Observation 6.2.10 to get
b(1/2)kgk .× dG (id, g).
Letting C > 0 be the implied constant, we have
NX,G (ρ) ≥ NG (bρ/2 /C) ∀ρ > 0

(11.2.6)

(cf. Remark 8.1.3). In particular, by (8.1.2)
δG = lim

ρ→∞

logb NG (R)
αG
logb NX,G (ρ)
≥ lim
=
·
R→∞ 2 logb (R)
ρ
2

To demonstrate the final assertion of Theorem 11.2.6, suppose that equality holds
in (11.2.5) and that δG < ∞. Then by Theorem 11.2.5, G is virtually nilpotent. Combining (11.2.6) with (11.2.2) and then plugging into (8.1.1) gives us that
Σδ (G) = ∞, completing the proof.



11.2.3. Examples with explicit Poincaré exponents. Theorem 11.2.6
raises a natural question: do the exponents allowed by this theorem actually occur
as the Poincaré exponent of some parabolic group? More precisely, given a finitely
generated abstract group Γ and a number δ ≥ αΓ /2, does there exist a hyperbolic metric space X and an injective homomorphism Φ : Γ → Isom(X) such that

G = Φ(Γ) is a parabolic group satisfying δG = δ? If δ = αΓ /2, then the problem
appears to be difficult; cf. Remark 11.2.12. However, we can provide a complete
answer when δ > αΓ /2 by embedding Γ into Isom(B) and then using Poincaré
extension to get an embedding into Isom(E∞ ). Specifically, we have the following:
Theorem 11.2.11. Let Γ be a virtually nilpotent group, and let α = αΓ be the
growth rate of NΓ . Then for all δ > αΓ /2, there exists an injective homomorphism
Φ : Γ → Isom(B) such that
δ(Φ(Γ)) = δ.

Moreover, Φ(Γ) may be taken to be either of convergence type or of divergence type.
Remark 11.2.12. Theorem 11.2.11 raises the question of whether there exists
an injective homomorphism Φ : Γ → Isom(B) such that
(11.2.7)

δ(Φ(Γ)) = αΓ /2.

11.2. THE POINCARÉ EXPONENT OF A FINITELY GENERATED PARABOLIC GROUP187

It is readily computed that if the map γ 7→ Φ(γ)(0) is bi-Lipschitz, then (11.2.7)
holds. In particular, if Γ = Zd for some d ∈ N, then such a Φ is given by
Φ(n)(x) = x + (n, 0). By contrast, if Γ is a virtually nilpotent group which is

not virtually abelian, then it is known [59, Theorem 1.3] that there is no quasiisometric embedding ι : Γ → B (see also [145, Theorem A] for an earlier version
of this result which applied to nilpotent Lie groups). In particular, there is no homomorphism Φ : Γ → Isom(B) such that γ 7→ Φ(γ)(0) is a bi-Lipschitz embedding.

So this approach of constructing an injective homomorphism Φ satisfying (11.2.7)
is doomed to failure. However, it is possible that another approach will work. We
leave the question as an open problem.
Remark 11.2.13. Letting Γ = Z in Theorem 11.2.11, we have the following
corollary: For any δ > 1/2, there exists an isometry gδ ∈ Isom(B) such that the
cyclic group Gδ = (gδ )Z satisfies δ(Gδ ) = δ, and may be taken to be either of
convergence type or of divergence type. The isometries (gδ )δ>1/2 exhibit “intermediate” behavior between the isometry g1/2 (x) = x + e1 (which has Poincaré
exponent 1/2 as noted above) and the isometries described in the Edelstein-type
isometries 11.1.12, 11.1.14, and 11.1.18: since δ > 1/2, the sequence (gδn (0))∞
1 conn
∞
verges to infinity much more slowly than the sequence (g1/2 (0))1 , but since δ < ∞,

the sequence converges faster than in Example 11.1.18, not to mention Examples
11.1.12 and 11.1.14 where the sequence (gδn (0))∞
1 does not converge to infinity at
all (although it converges along a subsequence).
Remark 11.2.14. Theorem 11.2.11 leaves open the question of whether there

is a homomorphism Φ : Γ → Isom(B) such that Φ(Γ) is strongly discrete but
δ(Φ(Γ)) = ∞. If Γ = Z, this is answered affirmatively by Example 11.1.18, and if Γ

contains Z as a direct summand, i.e. Γ ≡ Z × Γ′ for some Γ′ ≤ Γ, then the answer
can be achieved by taking the direct sum of Example 11.1.18 with an arbitrary
strongly discrete homomorphism from Γ′ to Isom(B). However, the Heisenberg
group does not contain Z as a direct summand. Thus, it is unclear whether or not
there is a a homomorphism from the Heisenberg group to Isom(B) whose image is
strongly discrete with infinite Poincaré exponent.
Proof of Theorem 11.2.11. We will need the following variant of the Assouad embedding theorem:
Theorem 11.2.15. Let X be a doubling metric space,2 and let F : (0, ∞) →

(0, ∞) be a nondecreasing function such that
(11.2.8)

0 < α∗ (F ) ≤ α∗ (F ) < 1.

2Recall that a metric space X is doubling if there exists M > 0 such that for all x ∈ X and ρ > 0,
the ball B(x, ρ) can be covered by M balls of radius ρ/2.

188

11. PARABOLIC GROUPS

Here
log F (λR) − log F (R)
log(λ)
log F (λR) − log F (R)
α∗ (F ) := lim sup sup
·
log(λ)
λ→∞ R>0
α∗ (F ) := lim inf inf

λ→∞ R>0

Then there exist d ∈ N and a map ι : X → Rd such that for all x, y ∈ X,
kι(y) − ι(x)k ≍× F (d(x, y)).

(11.2.9)

Proof. The classical Assouad embedding theorem (see e.g. [91, Theorem
12.2]) gives the special case of Theorem 11.2.15 where F (t) = tε for some 0 < ε < 1.
It is possible to modify the standard proof of the classical version in order to
accomodate more general functions F satisfying (11.2.8); however, we prefer to
prove Theorem 11.2.15 directly as a consequence of the classical version.
Fix ε ∈ (α∗ (F ), 1), and let
F (s)
Fb(t) = tε inf ε ·
s≤t s

The inequality ε > α∗ (f ) implies that Fb ≍× F , so we may replace F by Fb without

affecting either the hypotheses or the conclusion of the theorem. Thus, we may
without loss of generality assume that the function t 7→ F (t)/tε is nonincreasing.
Let G(t) = F (t)1/ε , so that t 7→ G(t)/t is nonincreasing. It follows that
G(t + s) ≤ G(t) + G(s).
Combining with the fact that G is nondecreasing shows that G ◦ d is a metric on
X. On the other hand, since α∗ (G) = α∗ (F )/ε > 0, there exists λ > 0 such that
G(λt) ≥ 2G(t) for all t > 0. It follows that the metric G ◦ d is doubling. Thus we
may apply the classical Assouad embedding theorem to the metric space (X, G ◦ d)
and the function t 7→ tε , giving a map ι : X → Rd satisfying

kι(y) − ι(x)k ≍× Gε ◦ d(x, y) = F (d(x, y)).
This completes the proof.

⊳

Now let Γ be a virtually nilpotent group, and let dΓ be a Cayley metric on Γ.
Lemma 11.2.16. (Γ, dΓ ) is a doubling metric space.
Proof. For all γ ∈ Γ and R > 0, we have by Corollary 11.2.2
#(B(γ, R)) = #(γ(B(e, R))) = #(B(e, R)) ≍× (1 ∨ R)αΓ .
Now let S ⊆ B(γ, 2R) be a maximal R-separated set. Then {B(β, R) : β ∈ S} is a

cover of B(γ, 2R). On the other hand, {B(β, R/2) : β ∈ S} is a disjoint collection

11.2. THE POINCARÉ EXPONENT OF A FINITELY GENERATED PARABOLIC GROUP189

of subsets of B(γ, 3R), so
X

β∈S

#(S) ≤

#(B(β, R/2)) ≤ #(B(γ, 3R))

(1 ∨ 3R)αΓ
#(B(γ, 3R))
≍× 1,
≍×
minβ∈S #(B(β, R/2))
(1 ∨ R/2)αΓ

i.e. #(S) ≤ M for some M independent of γ and R. But then B(γ, 2R) can be

covered by M balls of radius R, proving that Γ is doubling.

⊳

Now let f : [1, ∞) → [1, ∞) be a continuous increasing function satisfying
α < α∗ (f ) ≤ α∗ (f ) < ∞

(11.2.10)
and f (1) = 1. Let

Then
0 < α∗ (F ) = min


f −1 (Rα ) R ≥ 1
F (R) =
.
R1/2
R≤1



1
α
,
2 α∗ (f )



≤ α∗ (F ) = max



1
α
,
2 α∗ (f )



< 1.

Thus F satisfies the hypotheses of Theorem 11.2.15, so there exists an embedding
ι : Γ → H satisfying (11.2.9). By [59, Proposition 4.4], we may without loss of

generality assume that ι(γ) = Φ(γ)(0) for some homomorphism Φ : Γ → Isom(B).
Now for all R ≥ 1,
NB,Φ(Γ) (R) = #{γ ∈ Γ : Dξ (o, Φ(γ)(o)) ≤ R}
= #{γ ∈ Γ : F (dΓ (e, γ)) ≤ R}
α
= NΓ (F −1 (R)) ≍× F −1 (R) = f (R).

In particular, given δ > αΓ /2 and k ∈ {0, 2}, we can let
f (R) = R2δ (1 + log(R))−k .

It is readily verified that α < α(f ) = 2δ < ∞, so in particular (11.2.10) holds.
By (8.1.2), δ(Φ(Γ)) = δ and by (8.1.1), Φ(Γ) is of divergence type if and only if
k = 0.



Remark 11.2.17. The above proof shows a little more that what was promised;
namely, it has been shown that
(i) for every function F : (0, ∞) → (0, ∞) satisfying (11.2.8), there exists an
injective homomorphism Φ : Γ → B such that kΦ(γ)(0)k ≍× F (d(e, γ))
for all γ ∈ Γ, and that

190

11. PARABOLIC GROUPS

(ii) for every function f : [1, ∞) → [1, ∞) satisfying (11.2.10), there exists a

group G ≤ Isom(B) isomorphic to Γ such that NB,G (R) ≍× f (R) for all
R ≥ 1.

The latter will be of particular interest in Chapter 17, in which the orbital counting
function of a parabolic subgroup of a geometrically finite group is shown to have
implications for the geometry of the Patterson–Sullivan measure via the Global
Measure Formula (Theorem 17.2.2).
We conclude this chapter by giving two examples of how the Poincaré exponents
of infinitely generated parabolic groups behave somewhat erratically.
Example 11.2.18 (A class of infinitely generated parabolic torsion groups).
Let (bn )∞
1 be an increasing sequence of positive real numbers, and for each n ∈ N,

let gn ∈ Isom(B) be the reflection across the hyperplane Hn := {x : xn = bn }.
Then G := hgn : n ∈ Ni is a strongly discrete subgroup of Isom(B) consisting
b is a strongly
of only torsion elements. It follows that its Poincaré extension G
discrete parabolic subgroup of Isom(H∞ ) with no parabolic element. To compute
b we use (11.1.1):
the Poincaré exponent of G,
Σs (G) =

X

g∈G

−2s

(1 ∨ kg(0)k)

=

X

1∨

S⊆N
finite

=

Y

X

1∨

S⊆N
finite

gn

n∈S

X

!

2

(2bn )

n∈S

(0)

!−s

!−2s

.

The special case bn = n gives
Σs (G) ≥

X

S⊆{1,...,N }

N
X

n=1

2

(2n)

!−s

≍× 2N N −3s −→ ∞ ∀s ≥ 0
N

and thus δ = ∞, while the special case bn = nn gives
Σs (G) ≤

∞
X

n=1

X

(nn )−2s =

∞
X

n=1

S⊆N
max(S)=n

2n−1 (nn )−2s < ∞ ∀s > 0

and thus δ = 0. Intermediate values of δ can be achieved either by setting bn =
2n/(2δ) , which gives a group of divergence type:
−s
∞
X
X
X
Σs (G) ≍×
≍×
1 ∨ max(2bn )2
b−2s
n
S⊆N
finite

n∈S

n=1

=

∞
X

n=1

S⊆N
max(S)=n

2n−1 2−ns/δ


= ∞
< ∞

for s ≤ δ
for s > δ

11.2. THE POINCARÉ EXPONENT OF A FINITELY GENERATED PARABOLIC GROUP191

or by setting bn = 2n/(2δ) n1/δ , which gives a group of convergence type:

∞
∞
= ∞ for s < δ
X
X
X
Σs (G) ≍×
b−2s
=
2n−1 2−ns/δ n−2s/δ
n
< ∞ for s ≥ δ
n=1
n=1
S⊆N
max(S)=n

Remark 11.2.19. In Example 11.2.18, for each n the hyperplane Hn is a totally
T
geodesic subset of E∞ which is invariant under G. However, the intersection n Hn
is trivial, since no point x ∈ bord E∞ \ {∞} can satisfy xn = bn for all n. In

particular, G does not act irreducibly on any nontrivial totally geodesic set S ⊆
bord H∞ .

Example 11.2.20 (A torsion-free infinitely generated parabolic group with finite Poincaré exponent). Let Γ = {n/2k : n ∈ Z, k ≥ 0}. Then Γ is an infinitely
generated abelian group. For each k ∈ N let Bk = k k , and define an action
Φ : Γ → Isom(ℓ2 (N; C)) by the following formula:
k

Φ(q)(x0 , x) = x0 + q, e2πi2 q (xk − Bk ) + Bk

 
,
k

i.e. Φ(q) is the direct sum of the Edelstein-type example (cf. Definition 11.1.11)
defined by the sequences ak = 2k q, bk = Bk with the map R ∋ x0 7→ x0 + q. It is
readily verified that Φ is a homomorphism (cf. (11.1.5)). We have
X
k
kΦ(q)(0)k2 = |q|2 +
Bk2 |e2πi2 q − 1|2
k

2

≍× |q| +

X

Bk2 d(2k q, Z)

k

≍× max(|q|2 , Bk2q ),
/ Z. Equivalently, kq is the unique
where kq is the largest integer such that 2kq q ∈
integer such that q = n/2kq +1 for some k.

192

11. PARABOLIC GROUPS

To compute the Poincaré exponent of G = Φ(Γ), fix s > 1/2 and observe that
X
Σs (G) =
(1 ∨ kg(0)k)−2s
g∈G

=

X
q∈Γ

≤
≍×
=

=
=
≍×

(|q| ∨ Bkq )−2s

XX

(|n|/2k+1 ∨ Bk )−2s

k∈N n∈Z
XZ ∞

k∈N

0

−2s
 x
dx
∨
B
k
2k+1

#
 x −2s
dx
dx +
k+1
0
2k+1 Bk 2
k∈N
"
#

∞
1−2s
X
k+1 1−2s
k+1 2s x
2
Bk
+ (2
)
1 − 2s x=2k+1 Bk
k∈N


X
1
2k+1 Bk1−2s
2k+1 Bk1−2s +
2s − 1
k∈N
X
X
2k Bk1−2s =
2k (k k )1−2s < ∞.
"
X Z

k∈N

2k+1 Bk

Bk−2s

Z

∞

k∈N

Thus δ(G) ≤ 1/2, but Theorem 11.2.6 guarantees that δ(G) ≥ δ(Φ(Z)) ≥ 1/2. So
δ(G) = 1/2.

CHAPTER 12

Geometrically finite and convex-cobounded groups
In this chapter we generalize the notion of geometrically finite groups to regularly geodesic strongly hyperbolic metric spaces, mainly CAT(-1) spaces. We
generalize finite-dimensional theorems such as the Beardon–Maskit theorem [21]
and Tukia’s isomorphism theorem [170, Theorem 3.3].
Standing Assumptions 12.0.21. Throughout this chapter, we assume that
(I) X is regularly geodesic and strongly hyperbolic, and that
(II) G ≤ Isom(X) is strongly discrete.
Recall that for x, y ∈ bord X, [x, y] denotes the geodesic segment, ray, or line
connecting x and y.
Note that we do not assume that G is nonelementary.

12.1. Some geometric shapes
To define geometrically finite groups requires three geometric concepts. The
first, the quasiconvex core Co of the group G, has already been introduced in Section
7.5. The remaining two concepts are horoballs and Dirichlet domains.
12.1.1. Horoballs.
Definition 12.1.1. A horoball is a set of the form
Hξ,t = {x ∈ X : B ξ (o, x) > t},
where ξ ∈ ∂X and t ∈ R. The point ξ is called the center of a horoball Hξ,t , and
will be denoted center(Hξ,t ). Note that for any horoball H, we have
H ∩ ∂X = {center(H)}.
(Cf. Figure 12.1.1.)
Lemma 12.1.2. For every horoball H ⊆ X, we have
Diam(H) ≍× b−d(o,H) .
193

194

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

ξ
Hξ,t

o

t

ξ
Hξ,t
t

o

Figure 12.1.1. Two pictures of the same horoball, in the ball
model and half-space model, respectively.
Proof. Write H = Hξ,t for some ξ ∈ ∂X, t ∈ R. If t < 0, then o ∈ H, so
d(o, H) = 0 and Diam(H) = 1. So suppose t ≥ 0. Then the intersection [o, ξ] ∩ ∂H

consists of a single point x0 satisfying kx0 k = t. It follows that d(o, H) ≤ kx0 k = t
and Diam(H) ≥ D(x0 , x0 ) = b−t . For the reverse directions, fix x ∈ H. Since

B ξ (o, x) > t, we have

kxk > t

and
D(x, ξ) = b−hx|ξio = b−[Bξ (o,x)+ho|ξix ] ≤ b− Bξ (o,x) < b−t .

It follows that Diam(H) ≍× b−t = b−d(o,H) .



Lemma 12.1.3 (Cf. Figure 12.1.2). Suppose that H is a horoball not containing
o. Then
Diam(H \ B(o, ρ)) ≤ 2e−(1/2)ρ .
Proof. Write H = Hξ,t for some ξ ∈ ∂X and t ∈ R; we have t ≥ 0 since
o∈
/ H. Then for all x ∈ H \ B(o, ρ),
hx|ξio =

1
1
1
[kxk + B ξ (o, x)] ≥ [ρ + t] ≥ ρ
2
2
2

and so D(x, ξ) ≤ e−(1/2)ρ .



12.1.2. Dirichlet domains.
Definition 12.1.4. Let G be a group acting by isometries on a metric space
X. Fix z ∈ X. We define the Dirichlet domain for G centered at z by
(12.1.1)
Dz := {x : d(z, x) ≤ d(z, g(x)) ∀g ∈ G} = {x : B x (z, g −1 (z)) ≤ 0 ∀g ∈ G}.

12.1. SOME GEOMETRIC SHAPES

ρ

195

ւH \ B(o, ρ)
ξ

o
H

Figure 12.1.2. The set H \ B(o, ρ) decreases in diameter as ρ → ∞.
The idea is that the Dirichlet domain is a “tile” whose iterates under G tile the
space X. This is made explicit in the following proposition:
Proposition 12.1.5. For all z ∈ X, G(Dz ) = X.
Proof. Fix x ∈ X. Since the group G is strongly discrete, the minimum
ming∈G {d(x, g(z))} is attained at some g ∈ G. Now for every h ∈ G, we have

d(x, g(z)) ≤ d(x, h(z)). Replacing h by gh, it follows that for every h ∈ G we have
d(x, g(z)) ≤ d(x, gh(z)) which is the same as d(g −1 (x), z) ≤ d(g −1 (x), h(z)). Thus
g −1 (x) ∈ Dz , i.e. x ∈ g(Dz ).



Corollary 12.1.6. Let S ⊆ X be a G-invariant set. The following are equiv-

alent:

(A) There exists a bounded set S0 ⊆ X such that S ⊆ G(S0 ).
(B) The set S ∩ Dz is bounded.

Proof of (A) ⇒ (B). Given x ∈ S ∩ Dz , fix g ∈ G with x ∈ g(S0 ). Then

d(z, x) ≤ d(z, g −1 (x)) ≍+ 0, i.e. x is in a bounded set.



Proof of (B) ⇒ (A). The set S0 = S ∩ Dz is such a set. Specifically, given
x ∈ S by Proposition 12.1.5 there exists g ∈ G such that x ∈ g(Dz ). Since S is

G-invariant, g −1 (x) ∈ S ∩ Dz = S0 .



Remark 12.1.7. It is tempting to define the Dirichlet domain of G centered at
z to be the set
Dz∗ := {x : d(z, x) < d(z, g(x)) ∀g ∈ G such that g(z) 6= z},

196

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

γ2

γ2

γ1−1

e

z

γ1

γ1−1

γ2−1

z

e

Dz∗

γ1

G(Dz∗ )

γ2−1

Figure 12.1.3. The Cayley graph of Γ = F2 (Z) = hγ1 , γ2 i. The
closure of the naive Dirichlet domain Dz∗ is the geodesic segment
Dz∗ = [e, γ1 ]. Its orbit G(Dz∗ ) is the union of all geodesic segments
which appear as horizontal lines in this picture.
and then to try to prove that G(Dz∗ ) = X. However, there is a simple example
which disproves this hypothesis. Let X be the Cayley graph of Γ = F2 (Z) =
hγ1 , γ2 i, let Φ : Γ → Isom(X) be the natural action, and let G = Φ(Γ). If we let
z = ((e, γ1 ), 1/2), then Dz∗ = {((e, γ1 ), t) : t ∈ (0, 1)}, and
G(Dz∗ ) = {((g, gγ1 ), t) : g ∈ Γ, t ∈ [0, 1]}.
This set excludes all elements of the form ((g, gγ2 ), t), t ∈ (0, 1). (Cf. Figure
12.1.3.)

Remark 12.1.8. The assumption that G is strongly discrete is crucial for
Proposition 12.1.5. In general, tiling Hilbert spaces turns out to be a very subtle problem and has been studied (among others) by Klee [113, 114], Fonf and
Lindenstrauss [75] and most recently by Preiss [146].

12.2. Cobounded and convex-cobounded groups
Before studying geometrically finite groups, we begin by considering the simpler
case of cobounded and convex-cobounded groups. The theory of these groups will
provide motivation for the theory of geometrically finite groups.
Definition 12.2.1. Let G be a group acting by isometries on a metric space
X. We say that G is cobounded if there exists σ > 0 such that X = G(B(o, σ)).
It has been a long-standing conjecture to prove or disprove the existence of
cobounded subgroups of Isom(H∞ ) that are discrete in an appropriate sense. To

12.2. COBOUNDED AND CONVEX-COBOUNDED GROUPS

197

the best of our knowledge, this conjecture was first stated explicitly by D. P. Sullivan
in his IHÉS seminar on conformal dynamics [164, p.17]. We give here two partial
answers to this question, both negative. Our first partial answer is as follows:
Proposition 12.2.2. A strongly discrete subgroup of Isom(H∞ ) cannot be
cobounded.
Proof. Let us work in the ball model B∞ . Suppose that G ≤ Isom(B∞ ) is a
strongly discrete cobounded group, and choose σ > 0 so that B∞ = G(BB (0, σ)).
Since G is strongly discrete, we have #(F ) < ∞ where
F := {x ∈ G(0) : dB (0, x) ≤ 2σ + 1}.
Choose v ∈ ∂B∞ such that BE (v, z) = 0 for all z ∈ F , and let x = tv, where
0 < t < 1 is chosen to make
dB (0, x) = σ + 1.
Since x ∈ B, we have x ∈ BB (y, σ) for some y ∈ G(0). But then d(0, y) ≤ 2σ + 1,
which implies y ∈ F , and thus BE (x, y) = 0. On the other hand
dB (x, y) ≤ σ < σ + 1 = dB (0, x),
which contradicts (2.5.1).



Proposition 12.2.2 leaves open the question of whether there exist cobounded
subgroups of Isom(H) which satisfy a weaker discreteness condition than strong
discreteness. One way that we could try to construct such a group would be to
take the direct limit of a sequence cobounded subgroups of Isom(Hd ) as d → ∞.
The most promising candidate for such a direct limit has been the direct limit
of a sequence of arithmetic cocompact subgroups of Isom(Hd ). (See e.g. [23] for
the definition of an arithmetic subgroup of Isom(Hd ).) Nevertheless, such innocent
hopes are dashed by the following result:
Proposition 12.2.3. If Gd ≤ Isom(Hd ) is a sequence of arithmetic subgroups,
then the codiameter of Gd tends to infinity, that is, there is no σ > 0 such that
Gd (B(o, σ)) = Hd for every d.
Proof. It is known [23, Corollary 3.3] that the covolume of Gd tends to infinity
superexponentially fast as d → ∞. On the other hand, the volume of B(o, σ) in Hd
tends to zero superexponentially fast. Indeed, it is equal to
Z σ
d/2
(2π /Γ(d/2))
sinhd−1 (r) dr ≍× π d/2 σ d−1 /Γ(d/2).
0

Thus, for sufficiently large d, the volume of B(o, σ) is less than the covolume of Gd ,
which implies that Gd (B(o, σ)) $ Hd .



198

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

Remark 12.2.4. Proposition 12.2.3 strongly suggests, but does not prove, that
it is impossible to get a cobounded subgroup of Isom(H∞ ) as the direct limit of
arithmetic subgroups of Isom(Hd ). One might ask whether one can get a cobounded
subgroup of Isom(H∞ ) as the direct limit of non-arithmetic subgroups of Isom(Hd );
the analogous known lower bounds on volume [1, 110] are insufficient to disprove
this. However, this approach seems unlikely to work, for two reasons: first of all,
the much worse lower bounds for the covolumes of non-arithmetic groups may just
be a failure of technique; there are no known examples of non-arithmetic groups
with volume lower than the bound which holds for arithmetic groups, and it is
conjectured that there are no such examples [23, p.9]. Second of all, even if such
groups exist, they are of no use to the problem unless an entire sequence of groups
may be found, each one of which is a subgroup of all its higher dimensional analogues. Such structure exists in the arithmetic case but it is unclear whether or not
it will also exist in the non-arithmetic case.
From Propositions 12.2.2 and 12.2.3, we see that the theory of cobounded
groups acting on H∞ will be rather limited. Consequently we focus on the weaker
condition of convex-coboundedness.
For the remainder of this chapter, we return to our standing assumption that
the group G is strongly discrete.
Definition 12.2.5. We say that G ≤ Isom(X) is convex-cobounded if its re-

striction to the quasiconvex core Co is cobounded, or equivalently if there exists
σ > 0 such that
Co ⊆ G(B(o, σ)).
We remark that whether or not G is convex-cobounded is independent of the
base point o (cf. Proposition 7.5.9).
From Proposition 7.5.3 we immediately deduce the following:
Observation 12.2.6. If X is an algebraic hyperbolic space and if G is nonelementary, then the following are equivalent:
(A) G is convex-cobounded.
(B) There exists σ > 0 such that CΛ ⊆ G(B(o, σ)).
In particular, when X is finite-dimensional, we see that the notion of convexcoboundedness coincides with the standard notion of convex-cocompactness.
12.2.1. Characterizations of convex-coboundedness. The property of
convex-coboundedness can be characterized in terms of the limit set. Precisely:
Theorem 12.2.7. The following are equivalent:
(A) G is convex-cobounded.

12.2. COBOUNDED AND CONVEX-COBOUNDED GROUPS

199

(B) G is of compact type and any of the following hold:
(B1) Λ(G) = Λur,σ (G) for some σ > 0.
(B2) Λ(G) = Λur (G).
(B3) Λ(G) = Λr (G).
(B4) Λ(G) = Λh (G).
Remark 12.2.8. (B1)-(B4) should be regarded as equivalent conditions which
also assume that G is of compact type, so that there are a total of 5 equivalent
conditions in this theorem.
The implications (B1) ⇒ (B2) ⇒ (B3) ⇒ (B4) follow immediately from the
definitions. We therefore proceed to prove (A) ⇒ (B1) and (B4) ⇒ (A).
Proof of (A) ⇒ (B1). The proof consists of two parts: showing that Λ(G) =
Λur,σ (G) for some σ > 0, and showing that Λ(G) is compact.
Proof that Λ(G) = Λur,σ (G) for some σ > 0. Fix ξ ∈ Λ(G), so that
[o, ξ] ⊆ Co ⊆ G(B(o, σ)).
For each n ∈ N, let xn = [o, ξ]n , so that xn → ξ and d(xn , xn+1 ) = 1. Then for
each n, there exists gn ∈ G satisfying d(gn (o), xn ) ≤ σ. Then
ho|ξign (o) ≤ ho|ξixn + σ = σ;
moreover,
d(gn (o), gn+1 (o)) ≤ d(xn , xn+1 ) + 2σ = 2σ + 1.
Thus the convergence gn (o) → ξ is (2σ + 1)-uniformly radial, so ξ ∈ Λur,2σ+1 (G).
⊳
Proof that G is of compact type. By contradiction, suppose that G is
not of compact type. Then Λ is a complete metric space which is not compact,
which implies that there exist ε > 0 and an infinite ε-separated set I ⊆ Λ. Fix ρ > 0
large to be determined. For each ξ ∈ I, let xξ = [o, ξ]ρ . Then xξ ∈ Co ⊆ G(B(o, σ)),
so there exists gξ ∈ G such that d(gξ (o), xξ ) ≤ σ.

Claim 12.2.9. For ρ sufficiently large, the function ξ 7→ gξ (o) is injective.
Proof. Fix ξ1 , ξ2 ∈ I distinct, and suppose gξ1 (o) = gξ2 (o). Then (cf. Figure

12.2.1) we have that

hξ1 |ξ2 io ≥ hx1 |x2 io =

1
[2ρ − d(x1 , x2 )] ≥ ρ − σ.
2

On the other hand, since I is ε-separated we have hξ1 |ξ2 io ≤ − log(ε). This is a

contradiction if ρ > σ − log(ε).

⊳

200

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

ξ1

ξ2

x1
2σ
σ

σ

x2

gξ (o)

o

Figure 12.2.1. If gξ1 (o) = gξ2 (o), then ξ1 and ξ2 must be close
to each other.

The strong discreteness of G therefore implies
#(I) ≤ #{g ∈ G : kgk ≤ ρ + σ} < ∞,
which is a contradiction since #(I) = ∞ by assumption.

⊳

This completes the proof of (A) ⇒ (B1).



Proof of (B4) ⇒ (A). We use the notation (7.5.2).
Lemma 12.2.10. Λh ∩ Do′ = .
(Lemma 12.2.10 is true even without assuming (B4); this fact will be used in
the proof of Theorem 12.4.5 below.)
Proof. By contradiction fix ξ ∈ Λh ∩ Do′ . Since ξ ∈ (Do )′ , (12.1.1) gives

B ξ (o, g(o)) ≤ 0 for all g ∈ G (cf. Lemma 3.4.22). But then ξ ∈
/ Λh , since by definition ξ ∈ Λh if and only if there exists a sequence (gn )∞
satisfying
B ξ (o, gn (o)) →
1
+∞.

⊳

Now by (B4) and Observation 7.5.12, we have (Co ∩ Do )′ ⊆ Λ ∩ Do′ = Λh ∩ Do′ ,

and so (Co ∩ Do )′ = . By (C) of Proposition 7.7.2, we get that Co ∩ Do is bounded,

and Corollary 12.1.6 finishes the proof.



The proof of Theorem 12.2.7 is now complete.
Remark 12.2.11. (B4) ⇒ (A) may also be deduced as a consequence of Theo-

rem 12.4.5(B3)⇒(A) below; cf. Remark 12.4.11. However, the above prove is much
shorter. Alternatively, the above proof may be viewed as the “skeleton” of the proof
of Theorem 12.4.5(B3)⇒(A), which is made more complicated by the presence of
parabolic points.

12.3. BOUNDED PARABOLIC POINTS

201

12.2.2. Consequences of convex-coboundedness. The notion of convexcoboundedness also has several important consequences. In the following theorem,
G is endowed with an arbitrary Cayley metric (cf. Example 3.1.2).
Theorem 12.2.12 (Cf. [39, Proposition I.8.19]). Suppose that G is convexcobounded. Then:
(i) G is finitely generated.
(ii) The orbit map g 7→ g(o) is a quasi-isometric embedding (cf. Definition

3.3.9).
(iii) δG < ∞.

We shall prove Theorem 12.2.12 as a corollary of a similar statement about
geometrically finite groups; cf. Theorem 12.4.14 and Observation 12.4.15 below.
For now, we list some corollaries of Theorem 12.2.12.
Corollary 12.2.13. Suppose that G is convex-cobounded. Then G is wordhyperbolic, i.e. G is a hyperbolic metric space with respect to any Cayley metric.
Proof. This follows from Theorem 12.2.12(ii) and Theorem 3.3.10.



Corollary 12.2.14. Suppose that G is convex-cobounded. Then
dimH (Λ) = δ < ∞.
Proof. This follows from Theorem 12.2.12(iii), Theorem 1.2.1, and Theorem
12.2.7.


12.3. Bounded parabolic points
The difference between groups that are geometrically finite and those that
are convex-cobounded is the potential presence of bounded parabolic points in the
former. In the Standard Case, a parabolic fixed point ξ in the limit set of a geometrically finite group G, is said to be bounded if (Λ \ {ξ})/ Stab(G; ξ) is compact

[34, p.272]. We will have to modify this definition a bit to make it work for arbitrary hyperbolic metric spaces, but we show that in the usual case, our definition
coincides with the standard one (Remark 12.3.7).
Fix ξ ∈ ∂X. Recall that Eξ denotes the set bord X \ {ξ}.
Definition 12.3.1. A set S ⊆ Eξ is ξ-bounded if ξ ∈
/ S.
The motivation for this definition is that if X = Hd and ξ = ∞, then ξ-bounded
sets are exactly those which are bounded in the Euclidean metric. Actually, this
can be generalized as follows:

202

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

Observation 12.3.2. Fix S ⊆ Eξ . The following are equivalent:
(A) S is ξ-bounded.
(B) hx|ξio ≍+ 0 for all x ∈ X.
(C) Dξ (o, x) .× 1 for all x ∈ X.

(D) S has bounded diameter in the Dξ metametric.
Condition (D) motivates the terminology “ξ-bounded”.
Proof of Observation 12.3.2. (A) ⇔ (B) follows from the definition of the
topology on bord X, (B) ⇔ (C) follows from (3.6.6), and (C) ⇔ (D) is obvious. 
Now fix G ≤ Isom(X), and let Gξ denote the stabilizer of ξ relative to G.

Recall (Definition 6.2.7) that ξ is said to be a parabolic fixed point of G if Gξ is a
parabolic group, i.e. if Gξ (o) is unbounded and
g ∈ Gξ ⇒ g ′ (ξ) = 1.
(Here g ′ (ξ) is the dynamical derivative of g at ξ; cf. Proposition 4.2.12.)
Observation 12.3.3. If ξ is a parabolic point then ξ ∈ Λ.
Proof. This follows directly from Observation 6.2.11.



Definition 12.3.4. A parabolic point ξ ∈ Λ is a bounded parabolic point if

there exists a ξ-bounded set S ⊆ Eξ such that
(12.3.1)

G(o) ⊆ Gξ (S).

We denote the set of bounded parabolic points by Λbp .
Lemma 12.3.5. Let G ≤ Isom(X), and fix ξ ∈ ∂X. The following are equiva-

lent:

(A) ξ is a bounded parabolic point.
(B) All three of the following hold:
(BI) ξ ∈ Λ,
(BII) g ′ (ξ) = 1 ∀g ∈ Gξ , and

(BIII) there exists a ξ-bounded set S ⊆ Eξ satisfying (12.3.1).
Proof. The only thing to show is that if (B) holds, then Gξ (o) is unbounded.
By contradiction suppose otherwise. Let S be a ξ-bounded set satisfying (12.3.1).
Then for all x ∈ G(o), we have x ∈ h(S) for some h ∈ Gξ , and so
hx|ξio = hh−1 (x)|ξih−1 (o) ≍+ hh−1 (x)|ξio
≍+ 0.

(since Gξ (o) is bounded)
(since h−1 (x) ∈ S)

By Observation 12.3.2, the set G(o) is ξ-bounded and so ξ ∈
/ Λ, contradicting

(BI).



12.3. BOUNDED PARABOLIC POINTS

203

We now prove a lemma that summarizes a few geometric properties about
bounded parabolic points.
Lemma 12.3.6. Let ξ be a parabolic limit point of G. The following are equivalent:
(A) ξ is a bounded parabolic point, i.e. there exists a ξ-bounded set S ⊆ Eξ
such that
(12.3.2)

G(o) ⊆ Gξ (S).

(B) There exists a ξ-bounded set S ⊆ Eξ ∩ ∂X such that
(12.3.3)

Λ \ {ξ} ⊆ Gξ (S).

Moreover, if H is a horoball centered at ξ satisfying G(o) ∩ H = , then (A)-(B)
are moreover equivalent to the following:

(C) There exists a ξ-bounded set S ⊆ Eξ such that
(12.3.4)

Co \ H ⊆ Gξ (S).

(D) There exists ρ > 0 such that
(12.3.5)

Co ∩ ∂H ⊆ Gξ (B(o, ρ)).

Remark 12.3.7. The equivalence of conditions (A) and (B) implies that in the
Standard Case, our definition of a bounded parabolic point coincides with the usual
one.
Proof of (A) ⇒ (B). This is immediate since Λ \ {ξ} ⊆ G(o)(1)e .

Here

N1,e (S) denotes the 1-thickening of S with respect to the Euclidean metametric
Dξ .

Proof of (B) ⇒ (A). If #(Λ) = 1, then G = Gξ and there is nothing to

prove. Otherwise, let η1 , η2 ∈ Λ be distinct points.
Let S be as in (12.3.3). Fix x = gx (o) ∈ G. Since hgx (η1 )|gx (η2 )igx (o) ≍+ 0,

Gromov’s inequality implies that there exists i = 1, 2 such that hgx (ηi )|ξix ≍+ 0.
By (12.3.3), there exists hx ∈ Gξ such that h−1
x gx (ηi ) ∈ S. We have
−1
≍+ 0.
hh−1
x gx (ηi )|ξio ≍+ hhx gx (ηi )|ξih−1
x (x)

By Proposition 4.3.1(i), this means that o and yx := hx−1 (x) are both within a
bounded distance of the geodesic line [h−1
x gx (ηi ), ξ]. Since one of these two points
must lie closer to ξ then the other, we have either
(12.3.6)

hyx |ξio ≍+ 0 or ho|ξiyx ≍+ 0.

204

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

By contradiction, let us suppose that there exists a sequence xn ∈ G(o) such that

Dξ (o, yxn ) → ∞. (If no such sequence exists, then for some N ∈ N the set S =
{y ∈ X : Dξ (o, y) ≤ N } is a ξ-bounded set satisfying (12.3.2).) For n sufficiently

large, the first case of (12.3.6) cannot hold, so the second case holds. It follows
that yn := yxn → ξ radially. So ξ is a radial limit point of G. In the remainder of
the proof, we show that this yields a contradiction.
By Proposition 4.3.1(i), for each n ∈ N there exists a point zn ∈ [o, ξ] satisfying
d(yn , zn ) ≍+ 0.

(12.3.7)

Now let ρ be the implied constant of (12.3.7), and let δ be the implied constant
of Proposition 4.3.1(ii). Since G is strongly discrete, M := #{g ∈ G : kgk ≤
2ρ + 2δ} < ∞. Let F ⊆ Gξ be a finite set with cardinality strictly greater than M .
By Proposition 4.3.1(ii), there exists t > 0 such that for all y ∈ [o, ξ] with y > t,
then d(y, [h(o), ξ]) ≤ δ for all h ∈ F .

Suppose zn > t. Then for all h ∈ F , we have d(zn , [h(o), ξ]) ≤ δ. On the other

hand, h(zn ) ∈ [h(o), ξ] and B ξ (zn , h(zn )) = 0; this implies that d(zn , h(zn )) ≤ 2δ
and thus d(yn , h(yn )) ≤ 2ρ + 2δ. But yn = gn (o) for some gn ∈ G, so we have

kgn−1 hgn k ≤ 2ρ + 2δ. But since #(F ) > M , this contradicts the definition of M .
It follows that zn ≤ t. But then kyn k ≤ kzn k + ρ ≤ t + ρ, implying that the
sequence (yn )∞
1 is bounded, a contradiction.



For the remainder of the proof, we fix a horoball H = Hξ,t ⊆ X disjoint from

G(o).

Proof of (A) ⇒ (C). Let S be as in (12.3.2). Fix x ∈ Co \ H. Then there
exist g1 , g2 ∈ G with x ∈ [g1 (o), g2 (o)]. We have hg1 (o)|g2 (o)ix = 0, so by Gromov’s

inequality there exists i = 1, 2 such that hgi (o)|ξix ≍+ 0. By (3.6.6), we have
Dξ,x (x, gi (o)) ≍× 1, and combining with (4.2.6) gives
Dξ (x, gi (o)) ≍× eBξ (o,x) ≤ et ≍×,H 1.
Now by (12.3.2), there exists h ∈ Gξ such that h−1 (gi (o)) ∈ S. Then by Observation

6.2.9,

Dξ (o, h−1 (x)) ≤ Dξ (o, h−1 (gi (o))) + Dξ (x, gi (o)) .× 1.

Thus h−1 (x) lies in some ξ-bounded set which is independent of x.



Proof of (C) ⇒ (D). Let S be a ξ-bounded set satisfying (12.3.4). Then for
all x ∈ S ∩ ∂H, by (h) of Proposition 3.3.3 we have
kxk = 2

hx|ξio
| {z }

≍+ 0 since x∈S

−

B ξ (o, x)
| {z }

=t since x∈∂H

≍+,H 0.

12.3. BOUNDED PARABOLIC POINTS

205

ξ

H
x

h−1 (x)
o
h−1 g(o)

g(o)

Figure 12.3.1. By moving x close to o with respect to the d
metric, h−1 also moves g(o) close to o with respect to the Dξ
metametric.
Thus S ∩ ∂H ⊆ B(o, ρ) for sufficiently large ρ. Applying Gξ demonstrates (12.3.5).



Proof of (D) ⇒ (A). Let ρ be as in (12.3.5), and fix g ∈ G. Since by as-

sumption G(o) ∩ H = , we have B ξ (o, g(o)) ≤ t. Let x = [g(o), ξ]t−Bξ (o,g(o)) , so
that x ∈ [g(o), ξ] ∩ ∂H (cf. Figure 12.3.1). By (12.3.5), there exists h ∈ Gξ such

that x ∈ B(h(o), ρ). Then

hh−1 g(o)|ξio = hg(o)|ξih(o) ≤ hg(o)|ξix + d(h(o), x)
= d(h(o), x)

(since x ∈ [g(o), ξ])

≤ ρ.
This demonstrates that g(o) ∈ h(S) for some ξ-bounded set S.



Remark 12.3.8. The proof of (B) ⇒ (A) given above shows a little more than

asked for, namely that a parabolic point of a strongly discrete group cannot also
be a radial limit point.
It will also be useful to rephrase the above equivalent conditions in terms of a
Dirichlet domain of Gξ . Indeed, letting Do (Gξ ) denote such a Dirichlet domain, we
have the following analogue of Corollary 12.1.6:

Lemma 12.3.9. Let ξ be a parabolic point of G, and let S ⊆ Eξ be a Gξ -invariant

set. The following are equivalent:

(A) There exists a ξ-bounded set S0 ⊆ Eξ such that S ⊆ Gξ (S0 ).
(B) The set S ∩ Do (Gξ ) is ξ-bounded.

206

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

Proof. We first observe that for all x ∈ Eξ and h ∈ Gξ , (g) of Proposition

3.3.3 gives

hx|ξio − hx|ξih(o) =

1
1
[B x (o, h(o)) + B ξ (o, h(o))] = B x (o, h(o)).
2
2

In particular
x ∈ Do (Gξ ) ⇔ hx|ξio ≤ hx|ξih(o) ∀h ∈ Gξ
⇔ Dξ (x, ξ) ≤ Dξ (h(x), ξ) ∀h ∈ Gξ ,
i.e. Do (Gξ ) is the Dirichlet domain of o for the action of Gξ on the metametric

space (Eξ , Dξ ). Note that this action is isometric (Observation 6.2.9) and strongly
discrete (Proposition 7.7.4). Modifying the proof of Corollary 12.1.6 now yields the
conclusion.

Corollary 12.3.10. In Lemma 12.3.6, the equivalent conditions (A)-(D) are

also equivalent to:
(A′ ) G(o) ∩ Do (Gξ ) is ξ-bounded.

(B′ ) Do (Gξ ) ∩ Λ \ {ξ} is ξ-bounded.
(C′ ) Co ∩ Do (Gξ ) \ H is ξ-bounded.
12.4. Geometrically finite groups

Definition 12.4.1. We say that G is geometrically finite if there exists a disS
joint G-invariant collection of horoballs H satisfying o ∈
/ H such that
(I) for every ρ > 0, the set

Hρ := {H ∈ H : d(o, H) ≤ ρ}

(12.4.1)
is finite, and

(II) there exists σ > 0 such that
Co ⊆ G(B(o, σ)) ∪

(12.4.2)

[

H.

Observation 12.4.2. Notice that the following implications hold:
G cobounded ⇒ G convex-cobounded ⇒ G geometrically finite.
Indeed, G is convex-cobounded if and only if it satisfies Definition 12.4.1 with
H = .
Remark 12.4.3. It is not immediately obvious that the definition of geometrical
finiteness is independent of the basepoint o, but this follows from Theorems 12.4.5
and 12.4.14 below.

12.4. GEOMETRICALLY FINITE GROUPS

207

Remark 12.4.4. Geometrical finiteness is closely related to the notion of relative hyperbolicity of a group; see e.g. [37]. The main differences are:
1. Relative hyperbolicity is a property of an abstract group, whereas geometrical finiteness is a property of an isometric group action (equivalently, of
a subgroup of an isometry group)
2. The maximal parabolic subgroups of relatively hyperbolic groups are assumed to be finitely generated, whereas we do not make this assumption
(cf. Corollary 12.4.17(i)).
3. The relation between relative hyperbolicity and geometrical finiteness is
only available in retrospect, once one proves that both are equivalent
to a decomposition of the limit set into radial and bounded parabolic
limit points plus auxiliary assumptions (compare Theorem 12.4.5 with
[37, Definition 1]).
12.4.1. Characterizations of geometrical finiteness. We now state and
prove an analogue of Theorem 12.2.7 in the setting of geometrically finite groups.
In the Standard Case, the equivalence (A) ⇔ (B2) of the following theorem was

proven by A. F. Beardon and B. Maskit [21]. Note that while in Theorem 12.2.7,
one of the equivalent conditions involved the uniformly radial limit set, no such
characterization exists for geometrically finite groups. This is because for many
geometrically finite groups, the typical point on the limit set is neither parabolic
nor uniformly radial. (For example, the set of uniformly radial limit points of the
geometrically finite Fuchsian group SL2 (Z) is equal to the set of badly approximable
numbers; cf. e.g. [73, Observation 1.15 and Proposition 1.21].)
Theorem 12.4.5 (Generalization of the Beardon–Maskit Theorem; see also
[151, Proposition 1.10]). The following are equivalent:
(A) G is geometrically finite.
(B) G is of compact type and any of the following hold (cf. Remark 12.2.8):
(B1) Λ(G) = Λr,σ (G) ∪ Λbp (G) for some σ > 0.

(B2) Λ(G) = Λr (G) ∪ Λbp (G).
(B3) Λ(G) = Λh (G) ∪ Λbp (G).

Remark 12.4.6. Of the equivalent definitions of geometrical finiteness discussed in [34], it seems the above definitions most closely correspond with (GF1)
and (GF2).1 It seems that definitions (GF3) and (GF5) cannot be generalized to
our setting. Indeed, (GF5) depend on the notion of volume, which does not exist
in infinite dimensional spaces, while (GF3) already fails in the case of variable curvature; cf. [36]. It seems plausible that a version of (GF4) could be made to work
1Cf. Remark 12.3.7 above regarding (GF2).

208

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

at least in the setting of algebraic hyperbolic spaces, but we do not study the issue
at this stage.
The implications (B1) ⇒ (B2) ⇒ (B3) follow immediately from the definitions.

We therefore proceed to prove (A) ⇒ (B1) and then the more difficult (B3) ⇒ (A).

Proof of (A) ⇒ (B1). The proof consists of two parts: showing that Λ(G) =
Λr,σ (G) ∪ Λbp (G) for some σ > 0, and showing that G is of compact type.
Proof that Λ(G) = Λr,σ (G) ∪ Λbp (G) for some σ > 0. Let H be the disjoint G-invariant collection of horoballs as defined in Definition 12.4.1, and let
σ > 0 be large enough so that (12.4.2) holds. Fix ξ ∈ Λ, and we will show that

ξ ∈ Λr,σ ∪ Λbp . For each t ≥ 0, recall that [o, ξ]t denotes the unique point on [o, ξ]
so that d(o, [o, ξ]t ) = t; since [o, ξ]t ∈ Co , by (12.4.2) either [o, ξ]t ∈ G(B(o, σ)) or
S
[o, ξ]t ∈ H .
Now if there exists a sequence tn → ∞ satisfying [o, ξ]tn ∈ G(B(o, σ)), then
S
ξ ∈ Λr,σ (Corollary 4.5.5). Assume not; then there exists t0 such that [o, ξ]t ∈ H
for all t > t0 . This in turn implies that the collection

{t > t0 : [o, ξ]t ∈ H} : H ∈ H

is a disjoint open cover of (t0 , ∞). Since (t0 , ∞) is connected, we have (t0 , ∞) =

{t > t0 : [o, ξ]t ∈ H} for some H ∈ H , or equivalently
[o, ξ]t ∈ H ∀t > t0 .
Therefore ξ = center(H). Now it suffices to show

Lemma 12.4.7. For every H ∈ H , if center(H) ∈ Λ, then center(H) ∈ Λbp .
Proof. Let ξ = center(H). For every g ∈ Gξ , we have g(H) ∩ H 6= . Since

H is disjoint, this implies g(H) = H and thus g ′ (ξ) = 1. Thus ξ is neutral with
respect to every element of Gξ .
We will demonstrate equivalent condition (D) of Lemma 12.3.6. First of all, we
S
S
observe that G(o) is disjoint from H since o ∈
/ H . Fix x ∈ Co ∩ ∂H ⊆ Co \ H .
Then by (12.4.2), we have x ∈ gx (B(o, σ)) for some gx ∈ G. It follows that

gx−1 (x) ∈ B(o, σ) and so gx−1 (H) ∩ B(o, σ + ε) 6=  for every ε > 0. Equivalently,
gx−1 (H) ∈ Hσ+ε , where Hρ is defined as in (12.4.1). Therefore, by (I) of Definition
12.4.1, the set

{gx−1 (H) : x ∈ Co ∩ ∂H}

(H))n1 be an enumeration of this set. Then for any x ∈ Co ∩ ∂H
is finite. Let (gx−1
i

(H) = H and so
(H). Then gx gx−1
there exists i = 1, . . . , n with gx−1 (H) = gx−1
i
i

12.4. GEOMETRICALLY FINITE GROUPS

209

ξ1
x1

ξ2
x2

o

Hξi

Figure 12.4.1. If Hξ1 = Hξ2 , then ξ1 and ξ2 must be close to each other.
∈ Gξ . Thus
(ξ) = ξ. Equivalently, hx := gx gx−1
gx gx−1
i
i
n

k + kgx−1 (x)k ≤ σ + max kgxi k.
(o), gx−1 (x)) ≤ kgx−1
d(x, Gξ (o)) ≤ d(hx (o), x) = d(gx−1
i
i
i=1

Letting ρ = σ + maxni=1 kgxi k, we have (12.3.5), which completes the proof.

⊳

The identity Λ(G) = Λr,σ (G) ∪ Λbp (G) has been proven.

⊳

Proof that G is of compact type. By contradiction, suppose otherwise.
Then Λ is a complete metric space which is not compact, which implies that there
exist ε > 0 and an infinite ε-separated set I ⊆ Λ. Fix ρ > 0 large to be determined.
S
For each ξ ∈ I, let xξ = [o, ξ]ρ . Then xξ ∈ Co ⊆ G(B(o, σ)) ∪ H , so either
(1) there exists gξ ∈ G such that d(gξ (o), xξ ) ≤ σ, or
(2) there exists Hξ ∈ H such that xξ ∈ Hξ .

Claim 12.4.8. For ρ sufficiently large, the partial functions ξ 7→ gξ (o) and

ξ 7→ Hξ are injective.

Proof. For the first partial function ξ 7→ gξ (o), see Claim 12.2.9. Now fix
ξ1 , ξ2 ∈ I distinct, and suppose that Hξ1 = Hξ2 (cf. Figure 12.4.1). Then xi :=

xξi ∈ Hξi \ B(o, ρ). By Lemma 12.1.3, this implies that

ε ≤ D(ξ1 , ξ2 ) ≤ D(x1 , x2 ) ≤ 2e−(1/2)ρ .
For ρ > 2(log(2) − log(ε)), this is a contradiction. Thus the second partial function
ξ 7→ Hξ is also injective.

⊳

210

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

The strong discreteness of G and (12.4.1) therefore imply
#(I) ≤ #{H ∈ H : d(o, H) ≤ ρ} + #{g ∈ G : kgk ≤ ρ + σ} < ∞ ,
which is a contradiction since #(I) = ∞ by assumption.

⊳

This completes the proof of (A) ⇒ (B1).



Proof of (B3) ⇒ (A). Let F := (Co ∩Do )′ , where we use the notation (7.5.2).
By Lemma 12.2.10 Observation 7.5.12, and our hypothesis (B3), we have
F ⊆ Λ \ Λh ⊆ Λbp .

(12.4.3)
Claim 12.4.9. #(F ) < ∞.

Proof. Note that F is compact since G is of compact type and so it is enough
to show that F has no accumulation points. By contradiction, suppose there exists
ξ ∈ F such that ξ ∈ F \ {ξ}. Then by (12.4.3), ξ ∈ Λbp , so by (B′ ) of Corollary
12.3.10, Do (Gξ )∩Λ\{ξ} is ξ-bounded. But F \{ξ} ⊆ Do′ ∩Λ\{ξ} ⊆ Do (Gξ )∩Λ\{ξ},
contradicting that ξ ∈ F \ {ξ}.

⊳

Let P be a transversal of the partition of F into G-orbits. Fix t > 0 large to
be determined. For each p ∈ P let
Hp = Hp,t = {x : B p (o, x) > t},
and let
(12.4.4)

H := {g(Hp ) : p ∈ P, g ∈ G} .

Clearly, H is a G-invariant collection of horoballs. To finish the proof, we need to
show that:
S
(i) o ∈
/ H.
(ii) For t sufficiently large, H is a disjoint collection.
(iii) ((I) of Definition 12.4.1) For every ρ > 0 we have #(Hρ ) < ∞.

(iv) ((II) of Definition 12.4.1) There exists σ > 0 satisfying (12.4.2).

It turns out that (ii) is the hardest, so we prove it last.
Proof of (i). Fix g ∈ G and p ∈ P . Since p ∈ P ⊆ Do′ , we have
B p (o, g −1 (o)) ≤ 0 < t.
It follows that g −1 (o) ∈
/ Hp , or equivalently o ∈
/ g(Hp ).

⊳

Proof of (iii). Fix H = g(Hp ) ∈ Hσ for some p ∈ P . Consider the point

xH = [o, g(p)]d(o,H) ∈ ∂H, and note that d(o, xH ) = d(o, H) ≤ σ. Now g −1 (xH ) ∈

Hp , so by (D) of Lemma 12.3.6 there exists h ∈ Gp such that
d(h(o), g −1 (xH )) ≍+ 0.

12.4. GEOMETRICALLY FINITE GROUPS

211

p

g−1 (xH )

Hp

h−1 g−1 (xH )
o

Figure 12.4.2. Since g −1 (xH ) lies on the boundary of the
horoball Hp , an element of Gp can move it close to o.

(Cf. Figure 12.4.2.) Letting C be the implied constant, we have
kghk ≤ d(o, xH ) + d(xH , gh(o)) ≤ ρ + C.
On the other hand, gh(Hp ) = g(Hp ) = H since h ∈ Gp . Summarizing, we have
Hρ ⊆ {g(Hp ) : p ∈ P, kgk ≤ ρ + C}.
But this set is finite because G is strongly discrete and because of Claim 12.4.9.
Thus #(Hρ ) < ∞.
⊳
Proof of (iv).
Claim 12.4.10.


[ ′
Co ∩ Do \ H = .

Proof. By contradiction, suppose that there exists

[ ′
(12.4.5)
ξ ∈ Co ∩ Do \
H ⊆ F ⊆ Λbp .

By the definition of P , there exist p ∈ P and g ∈ G so that g(p) = ξ. Then
Hξ := g(Hp ) ∈ H is centered at ξ, and so by (C′ ) of Corollary 12.3.10,
Co ∩ Do \ Hξ ⊆ Do (Gξ ) ∩ Co \ Hξ
is ξ-bounded, contradicting (12.4.5).

⊳

S
Since G is of compact type, Claim 12.4.10 implies that the set Co ∩ Do \ H
is bounded (cf. (C) of Proposition 7.7.2), and Corollary 12.1.6 finishes the proof.
⊳

212

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

Proof of (ii). Fix H1 , H2 ∈ H distinct, and write Hi = gi (Hξi ) for i =

1, 2. The distinctness of H1 and H2 implies that they have different centers, i.e.
g1 (ξ1 ) 6= g2 (ξ2 ). (This is due to the inequivalence of distinct points in P .) By

contradiction, suppose that H1 ∩ H2 6= . Without loss of generality, we may
suppose that g1 = id and that g2 (ξ2 ) ∈ Do (Gξ1 ). Otherwise, let h ∈ Gξ1 be such

that hg1−1 g2 (ξ2 ) ∈ Do (Gξ1 ) (such an h exists by Proposition 12.1.5), and we have
Hξ1 ∩ hg1−1 g2 (Hξ2 ) 6= .
By (B′ ) of Corollary 12.3.10, we have

hξ1 |g2 (ξ2 )io ≍+ 0,
where the implied constant depends on ξ1 . Since there are only finitely many choices
for ξ1 , we may ignore this dependence.
Fix x ∈ H1 ∩ H2 . We have
B g2 (ξ2 ) (o, x) = B ξ2 (g2−1 (o), o) + B ξ2 (o, g2−1 (x))
> B ξ2 (g2−1 (o), o) + t

≥ 0 + t.

(since x ∈ H2 = g2 (Hξ2 ))

(since ξ2 ∈ Do′ )

On the other hand, B ξ1 (o, x) > t since x ∈ H1 . Thus (g) of Proposition 3.3.3 gives

1
B ξ1 (o, x) + B g2 (ξ2 ) (o, x)
2
1
≤ hξ1 |g2 (ξ2 )io − [t + t] ≍+ −t.
2
This is a contradiction for sufficiently large t.
0 ≤ hξ1 |g2 (ξ2 )ix = hξ1 |g2 (ξ2 )io −

The implication (B3) ⇒ (A) has been proven.

⊳


The proof of Theorem 12.4.5 is now complete.
Remark 12.4.11. The implication (B4) ⇒ (A) of Theorem 12.2.7 follows di-

rectly from the proof of the implication (B3) ⇒ (A) of Theorem 12.4.5, since if
there are no parabolic points then we have F =  and so no horoballs will be
defined in (12.4.4).

Observation 12.4.12. The proof of Theorem 12.4.5 shows that if G ≤ Isom(X)
is geometrically finite, then the set G\Λbp (G) is finite. When X = H3 , this is a
special case of Sullivan’s Cusp Finiteness Theorem [162], which applies to all finitely
generated subgroups of Isom(H3 ) (not just the geometrically finite ones). However,
the Cusp Finiteness Theorem does not generalize to higher dimensions [105].
Proof. Let H be the collection of horoballs defined in the proof of (B3)
⇒ (A), i.e. H = {g(Hp ) : p ∈ P } for some finite set P . We claim that Λbp = G(P ).
Indeed, fix ξ ∈ Λbp . By the proof of (A) ⇒ (B1), either ξ ∈ Λr or ξ = center(H)

12.4. GEOMETRICALLY FINITE GROUPS

213

for some H ∈ H . Since Λbp ∩ Λr =  (Remark 12.3.8), the latter possibility holds.
Write H = g(Hp ); then ξ = g(p) ∈ G(P ).



The set G\Λbp (G) is called the set of cusps of G.
Definition 12.4.13. A complete set of inequivalent parabolic points for a geometrically finite group G is a transversal of G\Λbp (G), i.e. a set P such that
Λbp = G(P ) but G(p1 ) ∩ G(p2 ) =  for all p1 , p2 ∈ P distinct.
Then Observation 12.4.12 can be interpreted as saying that any complete set
of inequivalent parabolic points for a geometrically finite group is finite.
12.4.2. Consequences of geometrical finiteness. Geometrical finiteness,
like convex-coboundedness, has some further geometric consequences. Recall (Theorem 12.2.12) that if G is convex-cobounded, then G is finitely generated, and for
any Cayley graph of G, the orbit map g 7→ g(o) is a quasi-isometric embedding. If
G is only geometrically finite rather than convex-cobounded, then in general neither of these things is true.2 Nevertheless, by considering a certain weighted Cayley
metric with infinitely many generators, we can recover the rough metric structure
of the orbit G(o).
Recall that the weighted Cayley metric of G with respect to a generating set
E0 and a weight function ℓ0 : E0 → (0, ∞) is the metric
dG (g1 , g2 ) :=

inf
n

n
X

(hi )1 ∈(E∪F )n
g1 =g2 h1 ···hn i=1

ℓ0 (hi ).

(Example 3.1.2). To describe the generating set and weight function that we want
to use, let P be a complete set of inequivalent parabolic points of G, and consider
the set
[
E :=
Gp .
p∈P

We will show that there exists a finite set F such that G is generated by E ∪ F .
Without loss of generality, we will assume that this set is symmetric, i.e. h−1 ∈ F

for all h ∈ F . For each h ∈ E ∪ F let
(12.4.6)

ℓ0 (h) := 1 ∨ khk.

We then claim that when G is endowed with its weighted Cayley metric with respect
to (E ∪ F, ℓ0 ), then the orbit map will be a quasi-isometric embedding. Specifically:
Theorem 12.4.14. If G is geometrically finite, then
2For examples of infinitely generated strongly discrete parabolic groups, see Examples 11.2.18 and

11.2.20; these examples can be extended to nonelementary examples by taking a Schottky product
with a lineal group. Theorem 11.2.6 guarantees that the orbit map of a parabolic group is never
a quasi-isometric embedding.

214

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

(i) There exists a finite set F such that G is generated by E ∪ F .

(ii) With the metric dG as above, the orbit map g 7→ g(o) is a quasi-isometric
embedding.
Observation 12.4.15. Theorem 12.2.12 follows directly from Theorem 12.4.14,
since by Theorem 12.2.7 we have Λbp =  if G is convex-cobounded.
We now begin the proof of Theorem 12.4.14. Of course, part (i) has been proven
already (Theorem 12.4.5).
Proof of (i) and (ii). Let H and σ be as in Definition 12.4.1. Without loss
of generality, we may suppose that H = {k(Hp,t ) : k ∈ G, p ∈ P } for some t > 0
(cf. the proof of Theorem 12.4.5).
Fix ρ > 2σ + 1 large to be determined, and let F = {g ∈ G : kgk ≤ ρ}. Then
F is finite since G is strongly discrete.
Claim 12.4.16. For all g ∈ G \ F , there exist h1 , h2 ∈ E ∪ F such that
kgk − d(h1 h2 (o), g(o)) &×,ρ 1 ∨ kh1 k ∨ kh2 k ≍× ℓ0 (h1 ) + ℓ0 (h2 ).
Proof. Let γ : [0, kgk] → [o, g(o)] be the unit speed parameterization. Let

I = [σ + 1, ρ − σ]. Then γ(I) ⊆ Co , so by (12.4.2), either γ(I) ∩ h(B(o, σ)) 6=  for
S
some h ∈ G, or γ(I) ⊆ H .
Case 1: γ(I) ∩ h(B(o, σ)) 6=  for some h ∈ G. In this case, fix x ∈ γ(I) ∩
h(B(o, σ)). Then

khk ≤ kxk + d(x, h(o)) ≤ (ρ − σ) + σ = ρ,
so h ∈ F . On the other hand,
d(h(o), g(o)) ≤ d(h(o), x) + d(x, g(o))
= d(h(o), x) + kgk − kxk
≤ σ + kgk − (σ + 1)
= kgk − 1,
so
kgk − d(h(o), g(o)) ≥ 1 ≍×,ρ khk.
The claim follows upon letting h1 = h and h2 = id.
S
Case 2: γ(I) ⊆ H . In this case, since γ(I) is connected and H is a disjoint
open cover of γ(I), there exists H ∈ H such that γ(I) ⊆ H. Since
γ(0), γ(kgk) ∈ G(o) ⊆ X \ H, there exist

0 < t1 < σ + 1 < ρ − σ < t2 < kgk
so that γ(t1 ), γ(t2 ) ∈ ∂H. Let xi = γ(ti ) for i = 1, 2 (cf. Figure 12.4.3).

12.4. GEOMETRICALLY FINITE GROUPS

215

k−1 (p)

p

γ(I)
x1

kj1 (o)
o

k−1 γ(I)

Hp

x2

k−1

kj2 (o)

j1 (o)

k−1 (x2 )

k−1 (x1 )

j2 (o)

k−1 (o)
g(o)

k−1 (Hp )

k

−1

g(o)

Figure 12.4.3. Since j1−1 j2 ∈ E and kj1 ∈ F , the points o, kj1 (o),
and kj2 (o) are connected to each other by edges in the weighted
Cayley graph. Since the distance from kj2 (o) to g(o) are both
significantly less than the distance from o to g(o), our recursive
algorithm will eventually halt.

Since H ∈ H , we have H = k(Hp ) for some p ∈ P and k ∈ G. By
(D) of Lemma 12.3.6, there exist j1 , j2 ∈ Gp with
d(k −1 (xi ), ji (o)) ≤ ρp (i = 1, 2)
for some ρp > 0 depending only on p. Letting ρ0 = maxp∈P ρp , we have
kkj1 k ≤ kx1 k + d(x1 , kj1 (o)) ≤ (σ + 1) + ρ0 .
Letting ρ = max(ρ0 + σ + 1, 2σ + 2), we see that kkj1 k ≤ ρ, so h1 := kj1 ∈

F . On the other hand, h2 := j1−1 j2 ∈ E by construction, since j1 , j2 ∈ Gp .
Observe that h1 h2 = kj2 . Now
d(h1 h2 (o), g(o)) ≤ d(g(o), x2 ) + d(x2 , kj2 (o))
≤ (kgk − t2 ) + ρ0 ,
and so
kgk − d(h1 h2 (o), g(o)) ≥ t2 − ρ0 .

(12.4.7)
Now

t2 ≥ t2 − t1 = d(x1 , x2 )

≥ d(j1 (o), j2 (o)) − d(k −1 (x1 ), j1 (o)) − d(k −1 (x2 ), j2 (o))

≥ kj1−1 j2 k − 2ρ0 = kh2 k − 2ρ0
and on the other hand

t2 ≥ ρ − σ ≥ ρ0 + 1.

216

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

Combining with (12.4.7), we see that
kgk − d(h1 h2 (o), g(o)) ≥ (kh2 k − 2ρ0 ) ∨ (ρ0 + 1) − ρ0
= (kh2 k − 3ρ0 ) ∨ 1
≍× 1 ∨ kh1 k ∨ kh2 k.
⊳

Fix j ∈ G, and define the sequence (hi )n1 in E ∪ F inductively as follows: If

h1 , . . . , h2i have been defined for some i ≥ 0, then let

−1
g = g2i = h2i
· · · h1−1 j = (h1 · · · h2i )−1 j.

(Note that g0 = j.) If g ∈ F , then let h2i+1 = g and let n = 2i + 1 (i.e. stop the
sequence here). Otherwise, by Claim 12.4.16 there exist h2i+1 , h2i+2 ∈ E ∪ F such
that

(12.4.8)

kg2i k − d(h2i+1 h2i+2 (o), g2i (o)) &×,ρ ℓ0 (h2i+1 ) + ℓ0 (h2i+2 ).

This completes the inductive step, as now h1 , . . . , h2(i+1) have been defined. We
remark that a priori, this process could be infinite and so we could have n = ∞;
however, it will soon be clear that n is always finite.
We observe that (12.4.8) may be rewritten:
kg2i k − kg2(i+1) k &×,ρ ℓ0 (h2i+1 ) + ℓ0 (h2i+2 ).
Iterating yields
(12.4.9)

kjk − kg2m k &×

2m
X
i=1

ℓ0 (hi ) ∀m ≤ n/2.

In particular, since ℓ0 (hi ) ≥ 1 for all i, we have
kjk &× ⌊n/2⌋ ≍× n,
and thus n < ∞. This demonstrates that the sequence (hi )n1 is in fact a finite

sequence. In particular, since the only way the sequence can terminate is if g2i ∈ F

for some i ≥ 0, we have gn−1 ∈ F and hn = gn−1 . From the definition of gn−1 ,
it follows that j = h1 · · · hn . Since j was arbitrary and h1 , . . . , hn ∈ E ∪ F , this
demonstrates that E ∪ F generates G, completing the proof of (i).

12.4. GEOMETRICALLY FINITE GROUPS

217

To demonstrate (ii), we observe that by (12.4.9) we have
kjk &×
≍+

n−1
X

i=1
n
X

ℓ0 (hi )
(since hn ∈ F )

ℓ0 (hi )

i=1

≥ dG (id, j),
where dG denotes the weighted Cayley metric. Conversely, if (hi )n1 is any sequence
satisfying j = h1 · · · hn , then
kjk ≤

n
X
i=1

d(h1 · · · hi−1 (o), h1 · · · hi (o)) =

and taking the infimum gives kjk ≤ dG (id, j).

n
X
i=1

khi k ≤

n
X

ℓ0 (hi ),

i=1



This finishes the proof of Theorem 12.4.14.

Corollary 12.4.17. If G is geometrically finite, then
(i) If for every ξ ∈ Λbp , Gξ is finitely generated, then G is finitely generated.
(ii) If for every ξ ∈ Λbp , δ(Gξ ) < ∞, then δ(G) < ∞.
Proof of (i). This is immediate from Theorem 12.4.14(i) and Observation
12.4.12.



Proof of (ii). Call a sequence (hi )n1 ∈ E n minimal if
(12.4.10)

n
X
i=1

ℓ0 (hi ) = dG (id, h1 · · · hn ).

Then for each g ∈ G \ {id}, there exists a minimal sequence (hi )n1 ∈ (E ∪ F )n so

that g = h1 · · · hn .
Let C be the implied multiplicative constant of (12.4.10), so that for every
minimal sequence (hi )n1 , we have
n
X
i=1

ℓ0 (hi ) &+

1
kh1 · · · hm k.
C

218

12. GEOMETRICALLY FINITE AND CONVEX-COBOUNDED GROUPS

Fix s > 0. Then
Σs (G) − 1 ≤

=

X

X

X

X

X

e−skgk

X

X

X

X

e−skgk

n
g∈G\{id} n∈N (hi )n
1 ∈(E∪F )
minimal
g=h1 ···hn

n
n∈N (hi )n
1 ∈(E∪F )
minimal

.×

n∈N (hi )n
1 ∈(E∪F )
minimal

≤
=

n∈N (hi )n
1 ∈(E∪F )

X

!
n
s X
exp −
ℓ0 (hi )
C i=1
n
!
n
s X
exp −
ℓ0 (hi )
C i=1
n
n
Y

X

e−(s/C)ℓ0 (hi )

n i=1
n∈N (hi )n
1 ∈(E∪F )

=

n
XY
X

e−(s/C)ℓ0 (h)

n∈N i=1 h∈E∪F

=

X

n∈N

In particular, if
λs :=

X

h∈E∪F

X

e

−(s/C)ℓ0 (h)

!n

.

e−(s/C)ℓ0 (h) < 1,

h∈E∪F

then Σs (G) < ∞. Now when s/C > maxp∈P δ(Gp ), we have λs < ∞. On the other

hand, each term of the sum defining λs tends to zero as s → ∞. Thus λs → 0 as
s → ∞, and in particular there exists some value of s for which λs < 1. For this s,

Σs (G) < ∞ and so δG ≤ s < ∞.



12.4.3. Examples of geometrically finite groups. We conclude this section by giving some basic examples of geometrically finite groups. We begin with
the following observation:
Observation 12.4.18.
(i) Any elliptic or lineal group is convex-cobounded.
(ii) Any parabolic group is geometrically finite and is not convex-cobounded.
Proof. This follows directly from Theorems 12.2.7 and 12.4.5. It may also be
proven directly; we leave this as an exercise to the reader.



Proposition 12.4.19. The strongly separated Schottky product G = hGa ia∈E

of a finite collection of geometrically finite groups is geometrically finite. Moreover,

12.4. GEOMETRICALLY FINITE GROUPS

219

if P1 and P2 are complete sets of inequivalent parabolic points for G1 and G2 respectively, then P = P1 ∪ P2 is a complete set of inequivalent parabolic points for
G. In particular, if the groups (Ga )a∈E are convex-cobounded, then G is convexcobounded.
Proof. This follows direction from Lemma 10.4.4, Theorem 10.4.7, Corollary
10.4.8, and Theorem 12.4.5.

Combining Observation 12.4.18 and Proposition 12.4.19 yields the following:
Corollary 12.4.20. The Schottky product of finitely many parabolic and/or
lineal groups is geometrically finite. If only lineal groups occur in the product, then
it is convex-cobounded.

CHAPTER 13

Counterexamples
In Chapter 5 we defined various notions of discreteness and demonstrated some
relations between them, and in Section 9.3 we related some of these notions to the
e In this chapter we give counterexamples to show
modified Poincaré exponent δ.

that the relations which we did not prove are in fact false. Specifically, we prove
that no more arrows can be added to Table 1 (reproduced below as Table 1), and
that the discreteness hypotheses of Proposition 9.3.1 cannot be weakened.

Finite dimensional
Riemannian manifold

General metric space
Infinite dimensional
algebraic hyperbolic space

Proper metric space

SD ↔ MD
↑
PrD
SD → MD
ր
PrD
SD → MD
ր
PrD
SD ↔ MD
↑
PrD

↔

WD
l
COTD ↔ UOTD
→
WD
ց
COTD
→
WD
↓
COTD → UOTD
↔ COTD
↓
WD

Table 1. The relations between different notions of discreteness.
COTD and UOTD stand for discrete with respect to the compactopen and uniform operator topologies respectively. All implications not listed have counterexamples, which are described below.

The examples are arranged roughly in order of discreteness level; the most
discrete examples are listed first.
We note that many of the examples below are examples of elementary groups.
In most cases, a nonelementary example can be achieved by taking the Schottky
product with an approprate group; cf. Proposition 10.5.1.
The notations B = ∂E∞ \ {∞} ≡ ℓ2 (N) and b· : Isom(B) → Isom(H∞ ) will be

used without comment; cf. Section 11.1.

221

222

13. COUNTEREXAMPLES

13.1. Embedding R-trees into real hyperbolic spaces
Many of the examples in this chapter are groups acting on R-trees, but it turns
out that there is a natural way to convert such an action into an action on a real
hyperbolic space. Specifically, we have the following:
Theorem 13.1.1 (Generalization of [40, Theorem 1.1]). Let X be a separable R-tree. Then for every λ > 1 there is an embedding Ψλ : X → H∞ and a
homomorphism πλ : Isom(X) → Isom(H∞ ) such that:
(i) The map Ψλ is πλ -equivariant and extends equivariantly to a boundary
map Ψλ : ∂X → ∂H∞ which is a homeomorphism onto its image.

(ii) For all x, y ∈ X we have
(13.1.1)

λd(x,y) = cosh d(Ψλ (x), Ψλ (y)).

(iii)
(13.1.2)

√
Hull1 (Ψλ (∂X)) ⊆ B(Ψλ (X), cosh−1 ( 2)).

(iv) For any set S ⊆ X, the dimension of the smallest totally geodesic subspace [VS ] ⊆ H∞ containing Ψλ (S) is #(S) − 1. Here cardinalities are
interpreted in the weak sense: if #(S) = ∞, then dim([VS ]) = ∞ but S

may be uncountable even though [VS ] is separable.

Proof. Let V = {x ∈ RX : xv = 0 for all but finitely many v ∈ X}, and

define the bilinear form BQ on V via the formula
X
(13.1.3)
BQ (x, y) = −
λd(v,w) xv yw .
v,w∈X

Claim 13.1.2. The associated quadratic form Q(x) = BQ (x, x) has signature

(ω, 1).

Proof. It suffices to show that Q ↿ e⊥
v0 is positive definite, where v0 ∈ X is

fixed. Indeed, fix x ∈ e⊥
v0 \ {0}, and we will show that Q(x) > 0. Now, the set
X0 = {v ∈ X : xv 6= 0} ∪ {v0 } is finite. It follows that the convex hull of X0 can

be written in the form X(V, E, ℓ) for some finite acyclic weighted undirected graph
(V, E, ℓ). Consider the subspace
V0 = {x ∈ e⊥
v0 : xv = 0 for all v ∈ X \ V } ⊆ V,
which contains x. We will construct an orthogonal basis for V0 as follows. For each
edge (v, w) ∈ E, let
fv,w = ev − λd(v,w) ew

13.1. EMBEDDING R-TREES INTO REAL HYPERBOLIC SPACES

223

if w ∈ [v0 , v]; otherwise let fv,w = fw,v . This vector has the following key property:
(13.1.4)

For all v ′ ∈ X, if [v, w] intersects [v0 , v ′ ] in
at most one point, then BQ (fv,w , ev′ ) = 0.

(The hypothesis implies that w ∈ [v ′ , v] and thus d(v, v ′ ) = d(v, w) + d(w, v ′ ).)
In particular, letting v ′ = v0 we see that fv,w ∈ e⊥
v0 . Moreover, the tree structure of (V, E) implies that for any two edges (v1 , w1 ) 6∼ (v2 , w2 ), we have either
#([v1 , w1 ] ∩ [v0 , v2 ]) ≤ 1 or #([v2 , w2 ] ∩ [v0 , v1 ]) ≤ 1; either way, (13.1.4) implies

that BQ (fv1 ,w1 , fv2 ,w2 ) = 0. Finally, Q(fv,w ) = λ2d(v,w) − 1 > 0 for all (v, w) ∈ E,
⊥
so Q ↿ V0 is positive definite. Thus Q(x) > 0; since x ∈ e⊥
v0 was arbitrary, Q ↿ ev0

is positive definite. This concludes the proof of the claim.

⊳

It follows that for any v ∈ X, the quadratic form
BQv (x, y) = BQ (x, y) + 2BQ (x, ev )BQ (ev , y)
is positive definite. We leave it as an exercise to show that for any v1 , v2 ∈ X,

the norms induced by Qv1 and Qv2 are comparable. Let L be the completion of
V with respect to any of these norms, and (abusing notation) let BQ denote the
unique continuous extension of BQ to L. Since the map X ∋ v 7→ ev ∈ L is

continuous with respect to the norms in question, L is separable. On the other
hand, since these norms are nondegenerate, we have dim(hev : v ∈ Si) = #(S)

for all S ⊆ X, and in particular dim(L) = ∞. Thus L is isomorphic to L∞ , so
H := {[x] ∈ P(L) : Q(x) < 0} is isomorphic to H∞ .

We define the embedding Ψλ : X → H via the formula Ψλ (v) = [ev ]. (13.1.1)
now follows immediately from (13.1.3) and (2.2.2). In particular, we have
d(Ψλ (v), Ψλ (w)) ≍+ log(λ)d(v, w),

which implies that Ψλ extends naturally to a boundary map Ψλ : ∂X → ∂Hα which
is a homeomorphism onto its image. Given any g ∈ Isom(X), we let πλ (g) = [Tg ] ∈
Isom(H), where Tg ∈ OR (L; Q) is given by the formula Tg (ev ) = eg(v) . Then Ψλ

and its extension are both πλ -equivariant, demonstrating condition (i).
For S ⊆ X, we have dim(VS ) = dim(hev : v ∈ Si) = #(S) as noted above, and
thus dim([VS ]) = dim(VS ) − 1 = #(S) − 1. This demonstrates (iv).
It remains to show (iii). Fix ξ, η ∈ ∂X and [z] ∈ [Ψλ (ξ), Ψλ (η)]. Write Ψλ (ξ) =

[x] and Ψλ (η) = [y]. Since [x], [y] ∈ ∂H and [z] ∈ H, we have Q(x) = Q(y) = 0,
and we may choose x, y, and z to satisfy BQ (x, y) = BQ (x, z) = BQ (y, z) = −1.
Since [z] ∈ [[x], [y]], we have z = ax + by for some a, b ≥ 0; we must have a = b = 1

and thus Q(z) = −2.

224

13. COUNTEREXAMPLES

Now, since Ψλ (w) = [ew ] → Ψλ (ξ) = [x] as w → ξ, there exists a function

f : X → R such that f (w)ew → x as w → ξ. Fixing v ∈ [ξ, η], we have
BQ (x, ev ) = lim f (w)BQ (ew , ev ) = − lim f (w)λd(v,w) .
w→ξ

w→ξ

In particular BQ (x, ev2 ) = BQ (x, ev1 )λBξ (v2 ,v1 ) , which implies that there exists
v ∈ [ξ, η] such that BQ (x, ev ) = 0. Similarly, there exists a function g : X → R
such that g(w′ )ew′ → y as w′ → η; we have

BQ (y, ev ) = − lim
g(w′ )λd(v,w
′

′

)

w →η

−1 = BQ (x, y) = lim f (w)g(w′ )BQ (ew , ew′ )
w→ξ
w ′ →η

′

= − lim f (w)g(w′ )λd(w,w ) = −BQ (x, ev )BQ (y, ev )
w→ξ
w ′ →η

BQ (x, ev ) = BQ (y, ev ) = −1,
so ev = z + w for some w ∈ x⊥ ∩ y⊥ . Since Q(z) = −2 and Q(ev ) = −1, we have
Q(w) = 1 and thus

√
2
|BQ (ev , z)|
= 2.
= √
cosh d([ev ], [z]) = p
1·2
|Q(ev )| · |Q(z)|
√
−1
In particular d([z], Ψλ (X)) ≤ cosh ( 2).



Definition 13.1.3. Given an R-tree X and a parameter λ > 1, the maps
Ψλ and πλ will be called the BIM embedding and the BIM representation with
parameter λ, respectively. (Here BIM stands for M. Burger, A. Iozzi, and N.
Monod, who proved the special case of Theorem 13.1.1 where X is an unweighted
simplicial tree.)
Remark 13.1.4. Let X, λ, Ψλ , and πλ be as in Theorem 13.1.1. Fix Γ ≤

Isom(X), and suppose that ΛΓ = ∂X. Let G = πλ (Γ) ≤ Isom(H∞ ).

(i) (13.1.2) implies that if Γ is convex-cobounded in the sense of Definition
12.2.5 below, then G is convex-cobounded as well. Moreover, we have
Λr (G) = ∂Ψλ (Λr (Γ)) and Λur (G) = ∂Ψλ (Λur (Γ)).
(ii) Since cosh(t) ≍× et for all t ≥ 0, (13.1.1) implies that
X
X
Σs (G) =
e−skπλ (γ)k ≍×
cosh−s (kπλ (γ)k)
γ∈Γ

γ∈Γ

=

X

γ∈Γ

λ−skγk = Σs log(λ) (Γ)

13.1. EMBEDDING R-TREES INTO REAL HYPERBOLIC SPACES

225

for all s ≥ 0. In particular δG = δΓ / log(λ). A similar argument shows
that δeG = δeΓ / log(λ), which implies that G is Poincaré regular if and only
if Γ is.

(iii) G is strongly discrete (resp. COT-discrete) if and only if Γ is strongly
discrete (resp. COT-discrete). However, this fails for weak discreteness;
cf. Example 13.4.2 below.
Proof of (iii). The difficult part is showing that if G is COT-discrete, then
Γ is as well. Suppose that Γ is not COT-discrete. Then there exists a sequence Γ ∋

γn → id in the compact-open topology. Let gn = πλ (γn ) ∈ G ≤ Isom(H∞ ) ≡ O(L).
Then the set
{x ∈ L : gn (x) → x}

contains Ψλ (X). On the other hand, since the sequence (gn )∞
1 is equicontinuous
(Lemma 2.4.11), this set is a closed linear subspace of L. Clearly, the only such
subspace which contains Ψλ (X) is L. Thus gn (x) → x for all x ∈ H∞ , and so
gn → id in the compact-open topology. Thus G is not COT-discrete.



We begin our list of examples with the following counterexample to an infinitedimensional analogue of Margulis’s lemma suggested in Remark 11.1.6:
Example 13.1.5. Let Γ = F2 (Z) = hγ1 , γ2 i, and let X be the Cayley graph of Γ.

Let Φ : Γ → Isom(X) be the natural action. Then H := Φ(Γ) is nonelementary and
strongly discrete. For each λ > 1, the image of H under the BIM representation πλ
is a nonelementary strongly discrete subgroup G = πλ (H) ≤ Isom(H∞ ) generated
by the elements g1 = πλ Φ(γ1 ), g2 = πλ Φ(γ2 ). But

cosh kgi k = λd(e,γi ) = λ,
so by an appropriate choice of λ, kgi k can be made arbitrarily small. So for arbitrarily small ε, we can find a free group G ≤ Isom(H∞ ) such that Gε (o) = G is nonelementary. This provides a counterexample to a hypothetical infinite-dimensional
analogue of Margulis’s lemma, namely, the claim that there exists ε > 0 such that
for every strongly discrete G ≤ Isom(H∞ ), Gε (o) is elementary.
Remark 13.1.6. If H is a finite-dimensional algebraic hyperbolic space and
G ≤ Isom(H) is nonelementary, then a theorem of I. Kim [112] states that the
length spectrum of G
L = {log g ′ (g− ) : g ∈ G is loxodromic}
is not contained in any discrete subgroup of R. Example 13.1.5 shows that this result
does not generalize to infinite-dimensional algebraic hyperbolic spaces. Indeed, if
G ≤ Isom(H∞ ) is as in Example 13.1.5 and if g = πλ (γ) ∈ G, then (13.1.1) implies

226

13. COUNTEREXAMPLES

that
log g ′ (g− ) = lim

n→∞

n
1 n
kg k = lim cosh−1 λkγ k
n→∞
n
1
= log(λ) lim kγ n k
n→∞ n
= log(λ) log γ ′ (γ− ),

demonstrating that L is contained in the discrete subgroup log(λ)Z ≤ R.
13.2. Strongly discrete groups with infinite Poincaré exponent
We have already seen two examples of strongly discrete groups with infinite
Poincaré exponent, namely the Edelstein-type Example 11.1.18, and the parabolic
torsion Example 11.2.18. We give three more examples here.
Example 13.2.1 (A nonelementary strongly discrete group G acting on a
proper R-tree X and satisfying δG = ∞). Let Y = [0, ∞), let P = N, and for
each p = n ∈ P let
Γp = Z/n!Z

(or more generally, let Γp be any sufficiently large finite group). Let (X, G) be the
geometric product of Y with (Γp )p∈P , as defined below in Example 14.5.10. By
Proposition 14.5.12, X is proper, and G = hGp ip∈P is a global weakly separated
Schottky product. So by Corollary 10.3.6, G is strongly discrete. Clearly, G is
nonelementary. Finally, δG = ∞ because for all s ≥ 0,
X
X
X X
e−skgk =
#(Γp \ {e})e−2skpk =
(n! − 1)e−2ns = ∞.
Σs (G) ≥
p∈P g∈Γp \{e}

p∈P

n∈N

Applying a BIM representation gives:
Example 13.2.2 (A nonelementary strongly discrete convex-cobounded group
acting on H∞ and satisfying δ = ∞). Cf. Remark 13.1.4 and the example above.
Example 13.2.3 (A parabolic strongly discrete group G acting on H∞ and
satisfying δG = ∞). Since F2 (Z) has the Haagerup property (Remark 11.1.2), there
is a homomorphism Φ : F2 (Z) → Isom(B) whose image G = Φ(F2 (Z)) is strongly
b must have infinite Poincaré exponent by Corollary 11.2.10.
discrete. However, G
13.3. Moderately discrete groups which are not strongly discrete

We have already seen one example of a moderately discrete group which is not
strongly discrete, namely the Edelstein-type Example 11.1.14 (parabolic acting on
H∞ ). We give three more examples here, and we will give one more example in
Section 13.4, namely Example 13.4.4. All five examples are are also examples of
properly discontinuous actions, so they also demonstrate that proper discontinuity

13.3. MODERATELY DISCRETE GROUPS WHICH ARE NOT STRONGLY DISCRETE 227

does not imply strong discreteness. (The fact that moderate discreteness (or even
strong discreteness) does not imply proper discontinuity can be seen e.g. from
Examples 11.2.18, 13.2.1, and 13.2.2, all of which are generated by torsion elements.)

H

∞

Example 13.3.1 (A parabolic group which acts properly discontinuously on
but is not strongly discrete). Let Z∞ ⊆ B = ℓ2 (N) denote the set of all infinite

sequences in Z with only finitely many nonzero entries. Let

G := {x 7→ x + n : n ∈ Z∞ } ⊆ Isom(B).
Then G acts properly discontinuously, since k(x + n) − xk ≥ 1 for all x ∈ B and
n ∈ Z∞ \ {0}. On the other hand, G is not strongly discrete since knk = 1 for
infinitely many n ∈ Z∞ . By Observation 11.1.1, these properties also hold for the
b ≤ Isom(H∞ ).
Poincaré extension G

Example 13.3.2 (A nonelementary group G which acts properly discontinuously on a separable R-tree X but is not strongly discrete). Let X be the Cayley

graph of Γ = F∞ (Z) with respect to its standard generators, and let Φ : Γ →
Isom(X) be the natural action. Then G = Φ(Γ) acts properly discontinuously on
X. On the other hand, since by definition each generator g ∈ G satisfies kgk = 1,
G is not strongly discrete.
Applying a BIM representation gives:
Example 13.3.3 (A nonelementary group which acts properly discontinuously
on H∞ but is not strongly discrete). Let X and G be as in Example 13.3.2. Fix λ >
1 large to be determined, and let πλ : Isom(X) → Isom(H∞ ) be the corresponding

BIM representation. By Remark 13.1.4, the group πλ (G) is a nonelementary group
which acts isometrically on H∞ but is not strongly discrete. To complete the proof,
we must show that πλ (G) acts properly discontinuously. By Proposition 10.4.10,
Q∞
it suffices to show that G = 1 πλ (γi )Z is a global strongly separated Schottky
group. And indeed, if we denote the generators of Γ = F∞ (Z) by γi (i ∈ N), and if
we consider the balls Ui± = B(Ψλ ((γi )± ), 1/2) (taken with respect to the Euclidean
metric), and if λ is sufficiently large, then the sets Ui = Ui+ ∪ Ui− form a global
strongly separated Schottky system for G.

Remark 13.3.4. The groups of Examples 13.3.2-13.3.3 can be easily modified
to make the group G uncountable at the cost of separability; let X be the Cayley
graph of F#(R) (Z) in Example 13.3.2, and applying (a modification of) Theorem
13.1.1 gives an action on H#(R) .
Remark. By Proposition 9.3.1, the groups of Examples 13.3.2-13.3.3 are all
Poincaré regular and therefore satisfy dimH (Λur ) = ∞.

228

13. COUNTEREXAMPLES

13.4. Poincaré irregular groups
We give six examples of Poincaré irregular groups, providing counterexamples
to many conceivable generalizations of Proposition 9.3.1.
Example 13.4.1 (A Poincaré irregular nonelementary group G acting on a
proper R-tree X which is weakly discrete but not COT-discrete). Let X be the
Cayley graph of V = F2 (Z) (equivalently, let X be the unique 3-regular unweighted
simplicial tree), and let G = Isom(X). Since #(Stab(G; e)) = ∞, G is not strongly
discrete, so by Proposition 5.2.7, G is also not COT-discrete. (The fact that G is
not COTD can also be deduced from Proposition 9.3.1, since we will soon show
that G is Poincaré irregular.)
On the other hand, suppose x ∈ X. Then either x ∈ V , or x = ((vx , wx ), tx )
for some (vx , wx ) ∈ E and tx ∈ (0, 1). In the first case, we observe that G(x) = V ,
while in the second we observe that
G(x) = {((v, w), tx ) : (v, w) ∈ E}.
In either case x is not an accumulation point of G(x). Thus G is weakly discrete.
To show that G is Poincaré irregular, we first observe that δ = ∞ since G is not
strongly discrete. On the other hand, Proposition 8.2.4(iv) can be used to compute
that δe = log (2). (Alternatively, one may use Theorem 1.2.3 together with the fact
b

that dimH (∂X) = logb (2).)

Remark. The group G in Example 13.4.1 is uncountable. However, if G is
replaced by a countable dense subgroup (cf. Remark 5.1.4) then the conclusions
stated above will not be affected. This remark applies also to Examples 13.4.2 and
13.4.4 below.
Applying a BIM representation to the group of Example 13.4.1 yields:
Example 13.4.2 (A Poincaré irregular nonelementary group acting irreducibly
on H∞ which is UOT-discrete but not COT-discrete). Let G ≤ Isom(X) be as in

Example 13.4.1 and let πλ : Isom(X) → Isom(H∞ ) ≡ O(L) be a BIM representation. Remark 13.1.4 shows that the group πλ (G) is Poincaré irregular and is not
COT-discrete. Note that it follows from either Proposition 5.2.7(ii) or Proposition
9.3.1 that πλ (G) is not weakly discrete, despite G being weakly discrete.
To complete the proof, we must show that πλ (G) is UOT-discrete. Let Ψλ :
X → H∞ ⊆ L be the BIM embedding corresponding to the BIM representation πλ ,

and write z = Ψλ (o); without loss of generality we may assume z = (1, 0), so that

Q(x) = kxk2 for all x ∈ z⊥ .
√
Now fix T = πλ (g) ∈ πλ (G)\{id}, and we will show that kT −Ik ≥ min( 2, λ−

1) > 0. We consider two cases. If g(o) 6= o, then kgk ≥ 1, which implies that

13.4. POINCARÉ IRREGULAR GROUPS

229

g(x)
y
o

x

Figure 13.4.1. The point y is the center of the triangle
∆(o, x, g(x)). Both o and y are fixed by g. Intuitively, this means
that g (really, πλ (g)) must have a significant rotational component
in order to “swing up” the point x to the point g(x).
|BQ (z, T z)| ≥ λ and thus that kT z − zk ≥ |BQ (z, T z − z)| ≥ λ − 1. So suppose

g(o) = o. Since g 6= id, we have g(x) 6= x for some x ∈ V ; choose such an x so as to
minimize kxk. Letting y = [o, x]kxk−1 , the minimality of kxk implies that g(y) = y
(cf. Figure 13.4.1).

Let x = Φλ (x) and y = Φλ (y), so that T y = y but T x 6= x. Let w1 = x − λy
and w2 = T w1 = T x − λy. An easy computation based on (13.1.1) and (2.2.2)
gives BQ (z, w1 ) = BQ (z, w2 ) = BQ (w1 , w2 ) = 0 (cf. (13.1.4)). It follows that
p
k(T − I)w1 k = kw2 − w1 k = Q(w2 − w1 )
p
= Q(w2 ) + Q(w1 )
p
√
= 2Q(w1 ) = 2kw1 k,
√
and thus kT − Ik ≥ 2.

Remark 13.4.3. Let G, πλ be as above and fix ξ ∈ ∂X. Then πλ (Gξ ) is a

focal group acting irreducibly on H∞ whose limit set is totally disconnected. This
contrasts with the finite-dimensional situation, where any nondiscrete group (and
thus any focal group) acting irreducibly on Hd is of the first kind [81, Theorem 2].
Example 13.4.4 (A Poincaré irregular nonelementary group G′ acting properly
discontinuously on a hyperbolic metric space X ′ ). Let G be the group described in
Example 13.4.1. Let X ′ = G and let

1 ∨ d(g(o), h(o))
d′ (g, h) :=
0

g 6= h

.

g=h

Since the orbit map X ′ ∋ g → g(o) ∈ X is a quasi-isometric embedding, (X ′ , d′ )

is a hyperbolic metric space. The left action of G on X ′ is isometric and properly
discontinuous. Denote its image in Isom(X ′ ) by G′ . Clearly δG′ = δG and δeG′ = δeG
(the Poincaré exponent and modified Poincaré exponent do not depend on whether
G is acting on X or on X ′ ), so G′ is Poincaré irregular.
The next set of examples have a somewhat different flavor.

230

13. COUNTEREXAMPLES

Example 13.4.5 (A Poincaré irregular group G acting on Hd ). Fix 2 ≤ d < ∞,

and let G be any nondiscrete subgroup of Isom(Hd ). Then

δeG = dimH (Λr ) ≤ dimH (∂Hd ) = d − 1.

On the other hand, since G is not strongly discrete we have δG = ∞. Thus G is
Poincaré irregular.
In Example 13.4.5, G could be a Lie subgroup with nontrivial connected component (e.g. G = Isom(Hd ), but this is not the only possibility - G can even be
finitely generated, as we now show:
Lemma 13.4.6. Let H be a connected algebraic group which contains a copy
of the free group F2 (Z). Then there exist g1 , g2 ∈ H such that G := hg1 , g2 i is a
nondiscrete group isomorphic to F2 (Z).
By Lemma 10.2.2, the group G cannot be a Schottky product - thus this lemma
provides an example of a free product which is not a Schottky product.
Proof. An orders-of-magnitude argument shows that there exists ε > 0 such
that for any h1 , h2 ∈ H with d(id, hi ) ≤ ε, we have
d(id, [h1 , h2 ]) ≤

1
max d(id, hi ),
2 i

where [h1 , h2 ] denotes the commutator of h1 and h2 . Thus for any g1 , g2 ∈ H such
that d(id, gi ) ≤ ε, letting

h1 = g1 , h2 = g2 , hn+2 = [hn , hn+1 ]
gives hn → id. But the elements hn are the images of nontrivial words in the free

group F2 (Z) under the natural homomorphism, so if this homomorphism is injective
then G is not discrete. For each element g ∈ F2 (Z), the set of homomorphisms
π : F2 (Z) → H such that π(g) = id is a proper algebraic subset of the set of all
homomorphisms, and therefore has measure zero. Thus for typical g1 , g2 satisfying

d(id, gi ) ≤ ε, G is a nondiscrete free group.



Instead of a Lie subgroup of Isom(Hd ), we could also take a locally compact
subgroup of Isom(H∞ ); there are many interesting examples of such subgroups. In
particular, one such example is given by the following theorem:
Theorem 13.4.7 (Monod–Py representation theorem, [132, Theorems B and
C]). For any d ∈ N and 0 < t < 1, there exist an irreducible representation ρt :

Isom(Hd ) → Isom(H∞ ) and a ρt -equivariant embedding ft : bord Hd → bord H∞
such that
(13.4.1)

d(ft (x), ft (y)) ≍+ td(x, y) for all x, y ∈ Hd .

13.4. POINCARÉ IRREGULAR GROUPS

231

The pair (ρt , ft ) is unique up to conjugacy.
Example 13.4.8 (A Poincaré irregular nonelementary group G acting irreducibly on H∞ ). Fix d ∈ N and 0 < t < 1, and let ρt , ft be as in Theorem 13.4.7.
Let Γ = Isom(Hd ), and let G = ρt (Γ). As G is locally compact, the modified
Poincaré exponent of G can be computed using Definition 8.2.1:


Z
−skgk
e
e
dg < ∞
δG = inf s ≥ 0 :
G


Z
e−skρt (γ)k dγ < ∞
= inf s ≥ 0 :
Γ


Z
= inf s ≥ 0 :
e−stkγk dγ < ∞
Γ

=

dimH (ΛΓ )
d−1
δeΓ
=
=
·
t
t
t

On the other hand, since G is convex-cobounded by [132, Theorem D], Theorem
12.2.12 shows that ΛG = Λr (G) = Λur (G). (It may be verified that the strong discreteness assumption is not needed for those directions.) Combining with Theorem
1.2.3, we have
d−1
> d − 1 = dimH (ΛΓ ).
t
In particular, it follows that the map ft : ΛΓ → ΛG cannot be smooth or even LipdimH (ΛG ) = dimH (Λr (G)) = dimH (Λur (G)) =

schitz. This contrasts with the smoothness of ft in the interior (see [132, Theorem
C(2)]).
Remark. The Hausdorff dimension of ΛG may also be computed directly from
the formulas (13.4.1) and (3.6.4), which imply that the map ft ↿ ΛΓ and its inverse
are Hölder continuous of exponents t and 1/t, respectively. However, the computation above gives a nice application of the Poincaré irregular case of Theorem
1.2.3.
In Examples 13.4.5 and 13.4.8, the group G does not satisfy any of the discreteness conditions discussed in Chapter 5. Our next example satisfies a weak
discreteness condition:
Example 13.4.9 (A Poincaré irregular nonelementary COT-discrete group G
acting reducibly on H∞ which is not weakly discrete). Let Γ = F2 (Z) and let
ι1 : Γ → Isom(Hd ) ≡ O(Ld+1 ) be an injective homomorphism whose image is a

nondiscrete group; this is possible by Lemma 13.4.6. Define ι2 : Γ → O(HΓ ) by
letting
ι2 (γ)[eδ ] = eγδ .
Note that ι2 (Γ) is COT-discrete, since kι2 (γ)ee − ee k =

√
2 for all γ ∈ Γ \ {e}.

232

13. COUNTEREXAMPLES

The direct sum ι := ι1 ⊕ ι2 : Γ → O(Ld+1 × HΓ ) is an isometric action of Γ on

˙
HΓ∪{1,...,d}
≡ H∞ . Let G = ι(Γ). Since ι1 (Γ) is the restriction of G to the invariant
totally geodesic subspace Hd , we have δG = δι1 (Γ) = ∞ and δeG = δeι1 (Γ) < ∞, so

G is Poincaré irregular. On the other hand, G is COT-discrete because ι2 (Γ) is.
Finally, the fact that G is not weakly discrete can be seen from either Observation
5.2.14 or Proposition 9.3.1.
13.5. Miscellaneous counterexamples
Our remaining examples include a COTD group which is not WD and a WD
group which is not MD.
Example 13.5.1 (A nonelementary COT-discrete group G which acts irreducibly on H∞ and satisfies δG = δeG = ∞ but which is not weakly discrete). Let
G1 ≤ Isom(H∞ ) be as in Example 13.4.9, and let g be a loxodromic isometry whose
˙
˙
fixed points are g± = [e0 ± ee ] ∈ ∂HΓ∪{1,...,d}
⊆ PLΓ∪{0,...,d}
. Then for n suffi-

ciently large, the product G = hG1 , (g n )Z i is a global strongly separated Schottky
product. By Lemma 10.2.2, G is COT-discrete. Since G contains G1 , G is not

weakly discrete.
The fact that δeG = ∞ follows from either Proposition 10.3.7(iii) or Proposition
9.3.1. So the only thing left to show is that G acts irreducibly. We assume that the

original group ι1 (Γ) acts irreducibly. Then if [V ] ⊆ H∞ is a G-invariant totally
geodesic subspace containing the limit set of G, then Ld+1 ⊆ V and so V =

Ld+1 ⊕ V2 for some V2 ⊆ HΓ . But [e0 + ee ] ∈ ΛG , so ee ∈ V2 . The G-invariance of
[V ] implies that V2 is ι2 (Γ)-invariant, and thus that V2 = HΓ and so [V ] = H∞ .
Remark. Example 13.5.1 gives a good example of how Theorem 1.2.3 gives
interesting information even when δeG = ∞. Namely, in this example Theorem 1.2.3
tells us that dimH (Λr ) = dimH (Λur ) = ∞, which is not at all obvious simply from
looking at the group.

Example 13.5.2 (An elliptic group G acting on H∞ which is weakly discrete
but not moderately discrete). Let H = ℓ2 (Z), and let T ∈ O(H) be the shift map
Z
T (x) = (xn+1 )∞
≤ O(H) ≤ Isom(B∞ ).
n=1 . Let G be the cyclic group G = T

Since g(0) = 0 for all g ∈ G, G is not moderately discrete. On the other hand, fix
x ∈ H \ {0}. Then T n (x) → 0 weakly as n → ±∞, so #{n ∈ Z : kT n(x) − xk ≤
kxk/2} < ∞. Thus G is weakly discrete.

CHAPTER 14

R-trees and their isometry groups
In this chapter we describe various ways to construct R-trees which admit
isometric actions. Section 14.1 describes the cone construction, in which one starts
with an ultrametric space (Z, D) and builds an R-tree X whose Gromov boundary
contains a point ∞ such that (Z, D) = (∂X \ {∞}, D∞,o). Sections 14.2 and 14.3
are preliminaries for Section 14.4, which describes the “stapling method” in which
one starts with a collection of R-trees (Xv )v∈V and staples them together to get

another R-tree. We give three very general examples of the stapling method in
which the resulting R-tree admits a natural isometric action.
We recall that whenever we have an example of an R-tree X with an isometric action Γ ≤ Isom(X), then we can get a corresponding example of a group of

isometries of H∞ by applying a BIM representation (Theorem 13.1.1). Thus, the
examples of this chapter contribute to our goal of understanding the behavior of
isometry groups acting on H∞ .

14.1. Construction of R-trees by the cone method
The construction of hyperbolic metric spaces by cone methods has a long history; see e.g. [85, 1.8.A.(b)], [168], [31, §7]. The construction below does not

appear to be equivalent to any of those existing in the literature, although our formula (14.1.1) is similar to [31, 7.1] (with the difference that their + sign is replaced
by a ∨; this change only works because we assume that Z is ultrametric).

Let (Z, D) be a complete ultrametric space. Define an equivalence relation
on Z × (0, ∞) by letting (z1 , r1 ) ∼ (z2 , r2 ) if d(z1 , z2 ) ≤ r1 = r2 , and denote the
equivalence class of (z, r) by hz, ri. Let X = Z × (0, ∞)/ ∼, and define a distance
function on X:

 2

r ∨ r22 ∨ D2 (z1 , z2 )
(14.1.1)
d hz1 , r1 i, hz2 , r2 i = log 1
r1 r2

(cf. Corollary 3.6.23). We call (X, d) the cone of (Z, D). Note that
(14.1.2)


(r0 ∨ r1 ∨ D(z0 , z1 ))(r0 ∨ r2 ∨ D(z0 , z2 ))
.
hz1 , r1 i|hz2 , r2 i hz0 ,r0 i = log
r0 (r1 ∨ r2 ∨ D(z1 , z2 ))
233

234

14. R-TREES AND THEIR ISOMETRY GROUPS

Theorem 14.1.1. The cone (X, d) is an R-tree. Moreover, there exists a map
ι : Z → ∂X such that ∂X\ι(Z) consists of one point, ∞, and such that D = D∞,o ◦ι,
where o = hz0 , 1i for any z0 ∈ Z.
Proof. Fix xi = hzi , ri i ∈ X, i = 1, 2, let R = r1 ∨ r2 ∨ D(z1 , z2 ), and let

γi : [log(ri ), log(R)] → X be defined by γi (t) = hzi , et i. Then γi parameterizes a
geodesic connecting xi and hzi , Ri. Since (z1 , R) ∼ (z2 , R), the geodesics γi can be

concatenated, and their concatenation is a geodesic connecting x1 and x2 . It can
be verified that the collection of such geodesics satisfies the conditions of Lemma
3.1.12. Thus (X, d) is an R-tree. (For an alternative proof that (X, d) is an R-tree,
see Example 14.5.1 below.)
Fix z0 ∈ Z. For all z1 , z2 ∈ Z and R > 0, (14.1.2) gives


2
X
√
D(z0 , zi )
√
− log D(z1 , z2 ).
log( R) ∨ log
lim hz1 , r1 i hz2 , r2 i hz0 ,Ri =
r1 ,r2 →0
R
i=1
∞
In particular, if z1 = z2 = z, then this shows that the sequence hz, 1/ni 1 is a

∞ 
∞
Gromov sequence. Let ι(z) = hz, 1/ni 1 . Similarly, the sequence hz0 , ni 1 is

∞ 
a Gromov sequence; let ∞ = hz0 , 1/ni 1 . Then Lemma 3.4.22 gives
hι(z1 )|ι(z2 )i(z0 ,R) =
and thus

2
X
i=1

− log D∞,o (ι(z1 ), ι(z2 )) = lim

log(R) ∨ log

R→∞

i.e. D∞,o ≡ D.



D(z0 , zi )
R



− log D(z1 , z2 )

h
i
hι(z1 )|ι(z2 )i(z0 ,R) − log(R) = − log D(z1 , z2 ),

To complete the proof we need to show that ∂X = ι(Z) ∪ {∞}. Indeed, fix

∞ 
ξ = hzn , rn i 1 ∈ ∂X. Without loss of generality suppose that rn → r ∈ [0, ∞]

and D(z0 , zn ) → R ∈ [0, ∞]. If r = ∞ or R = ∞, then it follows from (14.1.2) that
hhzn , rn i|∞ihz0 ,1i → ∞, i.e. ξ = ∞. Otherwise, it follows from (14.1.2) that
∞=

hzn , rn i|hzm , rm i hz0 ,1i


= 2 log(1 ∨ r ∨ R) − log
lim rn ∨ rm ∨ D(zn , zm ) ,
lim

n,m→∞

n,m→∞

which implies that rn ∨ rm ∨ D(zn , zm ) −−→ 0, i.e. rn → 0 and (zn )∞
1 is a Cauchy
n,m

sequence. Since Z is complete we can find a limit point zn → z ∈ Z. Then (14.1.2)

shows that ξ = ι(z).



Corollary 14.1.2. Every ultrametric space can be isometrically embedded into
an R-tree.

14.1. CONSTRUCTION OF R-TREES BY THE CONE METHOD

235

Proof. Let (Y, d) be an ultrametric space, and without loss of generality suppose that Y is complete. Let Z = Y , and let D(z1 , z2 ) = e(1/2)d(z1 ,z2 ) . Then (Z, D)
is a complete ultrametric space. Let (X, d) be the cone of (Z, D); by Theorem
14.1.1, X is an R-tree. Now define an embedding ι : Y → X via ι(y) = hy, 1i. Then
d(ι(y1 ), ι(y2 )) = 0 ∨ log D2 (y1 , y2 ) = d(y1 , y2 ),
i.e. ι is an isometric embedding.



Remark 14.1.3. Corollary 14.1.2 can also be proven from [31, Theorem 4.1]
by verifying directly that an ultrametric space satisfies Gromov’s inequality with
an implied constant of zero, and then proving that every geodesic metric space
satisfying Gromov’s inequality with an implied constant of zero is an R-tree.
However, the proof of Corollary 14.1.2 yields the additional information that
the isometric image of (Y, d) is contained in a horosphere, i.e.
B ∞ (ι(y1 ), ι(y2 )) = 0 ∀y1 , y2 ∈ Y,

(14.1.3)

where ∞ is as in Theorem 14.1.1.
Remark 14.1.4. The converse of the cone construction also holds: if (X, d) is
an R-tree and o ∈ X, ξ ∈ ∂X, then (∂X \ {ξ}, Dξ,o) and ({x ∈ X : B ξ (o, x) = 0}, d)
are both ultrametric spaces.

Proof. For all x, y ∈ Eξ , we have Dξ (x, y) = exp B ξ (o, C(x, y, ξ))), where
C(x, y, ξ) denotes the center of the geodesic triangle ∆(x, y, ξ) (cf. Definition
3.1.11). It can be verified by drawing appropriate diagrams (cf. Figure 3.3.1)
that for all x1 , x2 , x3 ∈ Eξ , there exists i such that C(xi , xj , ξ) = C(xi , xk , ξ) and

C(xj , xk , ξ) ∈ [ξ, C(xi , xj , ξ)] (where j, k are chosen so that {i, j, k} = {1, 2, 3}),
from which follows the ultrametric inequality for Dξ . Since Dξ = e(1/2)d on
{x ∈ X : B ξ (o, x) = 0}, the space ({x ∈ X : B ξ (o, x) = 0}, d) is also ultramet-

ric.



Theorem 14.1.5. Given an unbounded function f : [0, ∞) → N, the following
are equivalent:
(A) f is right-continuous and satisfies
∀R1 , R2 ≥ 0 such that R1 ≤ R2 , f (R1 ) divides f (R2 ).

(14.1.4)

(B) There exist an R-tree X (with a distinguished point o) and a parabolic
group G ≤ Isom(X) such that NX,G = f .

(C) There exist an R-tree X (with a distinguished point o) and a parabolic
group G ≤ Isom(X) such that NEp ,G = f , where p is the global fixed point
of G.

236

14. R-TREES AND THEIR ISOMETRY GROUPS

Moreover, in (B) and (C) the R-tree X may be chosen to be proper.
∞
Proof of (A) ⇒ (B). Let (λn )∞
1 and (Nn )1 be sequences such that
Y
f (ρ) =
Nn .
n∈N
λn ≤ρ

The hypotheses on f guarantee that (Nn )∞
1 can be chosen to be integers. Then for
each n ∈ N, let Γn be a finite group of cardinality Nn , and let
(
)
Y
∞
Γ = (γn )1 ∈
Γn : γn = e for all but finitely many n .
n∈N

For each

(γn )∞
1

∈ Γ let

(14.1.5)

k(γn )∞
1 k = max λn ,
n∈N
γn 6=e

with the understanding that kek = 0. For each α, β ∈ Γ let d(α, β) = kα−1 βk.

It is readily verified that d is an ultrametric on Γ. Thus by Corollary 14.1.2,
(Γ, d) can be isometrically embedded into an R-tree (X, d). Since Γ is proper, X
is proper. Moreover, the natural isometric action of Γ on itself extends naturally
to an isometric action on X. Denote this isometric action by φ, and let G = φ(Γ).
Then by (14.1.3), G is a parabolic group with global fixed point ∞. If we let o be
the image of e under the isometric embedding of Γ into X, then G satisfies
Y
NX,G (ρ) = #{γ ∈ Γ : kγk ≤ ρ} =
#(Γn ) = f (ρ).
n∈N
λn ≤ρ

This completes the proof.



Proof of (B) ⇒ (A). For each ρ > 0 let
Gρ = {g ∈ G : d(o, g(o)) ≤ ρ}.
Since G(o) is an ultrametric space by Remark 14.1.4, Gρ is a subgroup of G. Thus
by Lagrange’s theorem, the function f (ρ) = NX,G (ρ) = #(Gρ ) satisfies (14.1.4).
Since orbital counting functions are always right-continuous, this completes the
proof.



Proof of (A) ⇔ (C). Since the equation
NEp ,G (R) = NX,G (2 log(R))
holds for strongly hyperbolic spaces, including R-trees (Observation 6.2.10), and
since condition (A) is invariant under the transformation f 7→ (R 7→ f (2 log(R))),

the equivalence (A) ⇔ (B) directly implies the equivalence (A) ⇔ (C).



14.2. GRAPHS WITH CONTRACTIBLE CYCLES

237

Remark 14.1.6. Applying a BIM representation (Theorem 13.1.1) shows that
if f : [0, ∞) → N is an unbounded function satisfying (A) of Theorem 14.1.5, then
there exists a parabolic group G ≤ Isom(H∞ ) such that NX,G = f . This improves
a previous result of two of the authors [73, Proposition A.2].

14.2. Graphs with contractible cycles
In Section 14.4, we will describe a method of stapling together a collection of
R-trees (Xv )v∈V based on some data. This data will include a collection of edge
pairings E ⊆ V × V \ {(v, v) : v ∈ V } that indicates which trees are to be stapled to
each other. In this section, we describe the criterion which this collection of edge
pairings needs to satisfy in order for the construction to work (Definition 14.2.1),
and we analyze that criterion.
Let (V, E) be an unweighted undirected graph, and let dE denote the path
metric of (V, E) (cf. Definition 3.1.1). A sequence (vi )n0 in V will be called a path
if (vi , vi+1 ) ∈ E ∀i < n. The path (vi )n0 is said to connect the vertices v0 and vn .

The path (vi )n0 is called a geodesic if n = dE (v0 , vn ), in which case it is denoted
[v0 , vn ]. Note that a sequence is a geodesic if and only if [v0 , v1 ] ∗ · · · ∗ [vn−1 , vn ] is a

geodesic in the metrization X(V, E) (cf. Definition 3.1.1). Also, recall that a cycle
in (V, E) is a finite sequence of distinct vertices v1 , . . . , vn ∈ V , with n ≥ 3, such

that (v1 , v2 ), (v2 , v3 ), . . . , (vn−1 , vn ), (vn , v1 ) ∈ E (cf. (3.1.4)).

Definition 14.2.1. The graph (V, E) is said to have contractible cycles if every cycle forms a complete graph, i.e. if for every cycle (vi )n0 we have (vi , vj ) ∈
E ∀i, j such that vi 6= vj .
Standing Assumption 14.2.2. In the remainder of this section, (V, E) denotes
a connected graph with contractible cycles.
Lemma 14.2.3. For every v, w ∈ V there exists a unique geodesic [v, w] = (vi )n0

connecting v and w; moreover, if (wj )m
0 is any path connecting v and w, then the
vertices (vi )n0 appear in order (but not necessarily consecutively) in the sequence
(wj )m
0 .
Proof.
Claim 14.2.4. Let (vi )n0 be a geodesic, and let (wj )m
0 be a path connecting v0
and vn . Suppose n ≥ 2. Then there exist i = 1, . . . , n − 1 and j = 1, . . . , m − 1 such

that vi = wj .

Proof. By contradiction suppose not, and without loss of generality supm
pose that (wj )m
0 is minimal with this property. Then the vertices (wj )0 are distinct, since if we had wj1 = wj2 for some j1 < j2 , we could replace (wj )m
0 by

238

14. R-TREES AND THEIR ISOMETRY GROUPS

(w0 , . . . , wj1 −1 , wj1 = wj2 , wj2 +1 , . . . , wm ). Since n ≥ 2, it follows that the path

(v0 , v1 , . . . , vn = wm , wm−1 , . . . , w1 , w0 = v0 ) is a cycle. But then (v, w) ∈ E,
contradicting that (vi )n0 is a geodesic of length n ≥ 2.
⊳
Claim 14.2.5. Let (vi )n0 be a geodesic, and let (wj )m
0 be a path connecting v0
and vn . Then the vertices (vi )n0 appear in order in the sequence (wj )m
0 .

Proof. We proceed by induction on n. The cases n = 0, n = 1 are trivial.
Suppose the claim is true for all geodesics of length less than n. By Claim 14.2.4,
there exist i0 = 1, . . . , n − 1 and j0 = 1, . . . , m − 1 such that vi = wj . By the
induction hypothesis, the vertices (vi )i00 appear in order in the sequence (wj )j00 ,
and the vertices (vi )ni0 appear in order in the sequence (wj )m
j0 . Combining these
facts yields the conclusion.
⊳
To finish the proof of Lemma 14.2.3, it suffices to observe that if (vi )n0 and
(wj )m
0 are two geodesics connecting the same vertices v and w, then by Claim
14.2.5 the vertices (vi )n0 appear in order in the sequence (wj )m
0 , and the vertices
n
n
m
(wj )m
0 appear in order in the sequence (vi )0 . It follows that (vi )0 = (wj )0 , so
geodesics are unique.


Lemma 14.2.6 (Cf. Figure 14.2.1). Fix v1 , v2 , v3 ∈ V distinct. Then either
(1) there exists w ∈ V such that for all i 6= j, [vi , vj ] = [vi , w] ∗ [w, vj ], or

(2) there exists a cycle w1 , w2 , w3 ∈ V such that for all i 6= j, [vi , vj ] =
[vi , wi ] ∗ [wi , wj ] ∗ [wj , vj ].

Proof. For each i = 1, 2, 3, let ni be the number of initial vertices on which
the geodesics [vi , vj ] and [vi , vk ] agree, i.e.
ni = max{n : [vi , vj ]ℓ = [vi , vk ]ℓ ∀ℓ = 0, . . . , n},
and let wi = [vi , vj ]ni . Here j, k are chosen such that {i, j, k} = {1, 2, 3}. Then
uniqueness of geodesics implies that the geodesics [wi , wj ], i 6= j are disjoint ex-

cept for their common endpoints. If (wi )31 are distinct, then the path [w1 , w2 ] ∗
[w2 , w3 ] ∗ [w3 , w1 ] is a cycle, and since (V, E) has contractible cycles, this implies
(w1 , w2 ), (w2 , w3 ), (w3 , w1 ) ∈ E, completing the proof. Otherwise, we have wi = wj

for some i 6= j; letting w = wi = wj completes the proof.



Corollary 14.2.7 (Cf. Figure 14.2.2). Fix v1 , v2 , u ∈ V distinct such that
(v1 , v2 ) ∈ E. Then either v1 ∈ [u, v2 ], v2 ∈ [u, v1 ], or there exists w ∈ V such that

for each i = 1, 2, (w, vi ) ∈ E and w ∈ [u, vi ].

Proof. Write v3 = u, so that we can use the same notation as Lemma 14.2.6.
If we are in case (1), then the equation [v1 , v2 ] = [v1 , w] ∗ [w, v2 ] implies that

14.3. THE NEAREST-NEIGHBOR PROJECTION ONTO A CONVEX SET

v1

v1

w

w1

w2

239

w3

v2

v2

v3

v3

Figure 14.2.1. The two possibilities for a geodesic triangle in a
graph with contractible cycles. Lemma 14.2.6 states that either the
geodesic triangle looks like a triangle in an R-tree (right figure), or
there is 3-cycle in the “center” of the triangle (left figure).
v1
u

v1

v2

u

v2

v1

u

w
v2

Figure 14.2.2. When the vertices v1 and v2 are adjacent, Corollary 14.2.7 describes three possible pictures for the geodesic triangle ∆(u, v1 , v2 ). In the rightmost figure, w is the vertex adjacent
to both v1 and v2 from which the paths [u, v1 ] and [u, v2 ] diverge.
w ∈ {v1 , v2 }, and so either v1 = w ∈ [u, v2 ] or v2 = w ∈ [u, v1 ]. If we are in case

2, then the equation [v1 , v2 ] = [v1 , w1 ] ∗ [w1 , w2 ] ∗ [w2 , v2 ] implies that w1 = v1 and
w2 = v2 . Letting w = w3 completes the proof.

14.3. The nearest-neighbor projection onto a convex set
Let X be an R-tree, and let A ⊆ X be a nonempty closed convex set. Since

X is a CAT(-1) space, for each z ∈ X there is a unique point π(z) ∈ A such that

d(z, π(z)) = d(z, A), and the map z → π(z) is semicontracting (see e.g. [39]). Since
X is an R-tree, we can say more about this nearest-neighbor projection map π, as
well as providing a simpler proof of its existence. In the following theorems, X
denotes an R-tree.
Lemma 14.3.1. Let A ⊆ X be a nonempty closed convex set. Then for each

z ∈ X there exists a unique point π(z) ∈ A such that for all x ∈ A, π(z) ∈ [z, x].

240

14. R-TREES AND THEIR ISOMETRY GROUPS

Moreover, for all z1 , z2 ∈ X, we have
(14.3.1)

d(π(z1 ), π(z2 )) = 0 ∨ (d(z1 , z2 ) − d(z1 , A) − d(z2 , A)).

Proof. Since A is nonempty and closed, there exists a point π(z) ∈ A such

that [z, π(z)] ∩ A = {π(z)}. Fix z ∈ A. Since C(x, z, π(z)) ∈ [z, π(z)] ∩ [x, π(x)] ⊆
[z, π(z)] ∩ A, we get C(x, z, π(z)) = π(z), i.e. hx|ziπ(z) = 0, i.e. π(z) ∈ [z, x]. This
completes the proof of existence; uniqueness is trivial.
To demonstrate the equation (14.3.1), we consider two cases:

Case 1: If [z1 , z2 ] ∩ A 6= , then π(z1 ) and π(z2 ) both lie on the geodesic [z1 , z2 ],
so d(π(z1 ), π(z2 )) = d(z1 , z2 ) − d(z1 , A) − d(z2 , A) ≥ 0.

Case 2: Suppose that [z1 , z2 ] ∩ A = ; we claim that π(z1 ) = π(z2 ). Indeed, by

the definition of π(z2 ) we have π(z2 ) ∈ [z2 , π(z1 )], and by assumption we
have π(z2 ) ∈
/ [z1 , z2 ], so we must have π(z2 ) ∈ [z1 , π(z1 )]. But from the

definition of π(z1 ), this can only happen if π(z1 ) = π(z2 ). The proof is
completed by noting that the triangle inequality gives d(z1 , z2 )−d(z1 , A)−
d(z2 , A) = d(z1 , z2 ) − d(z1 , π(z1 )) − d(z2 , π(z1 )) ≤ 0.

Lemma 14.3.2. Let A1 , A2 ⊆ X be closed convex sets such that A1 ∩ A2 6= .

For each i let πi : X → Ai denote the nearest-neighbor projection map. Then for
all z ∈ X, either π1 (z) ∈ A2 or π2 (z) ∈ A1 . In particular, π1 (A2 ) ⊆ A1 ∩ A2 .

Proof. Let x1 = π1 (z) and x2 = π2 (z), and fix y ∈ A1 ∩ A2 . By Lemma
14.3.1, x1 , x2 ∈ [z, y]. Without loss of generality assume d(z, x1 ) ≤ d(z, x2 ), so that
x2 ∈ [x1 , y]. Since A1 is convex, x2 ∈ A1 .



Lemma 14.3.3. Let A1 , A2 ⊆ X be closed convex sets such that A1 ∩ A2 6= .

Then A1 ∪ A2 is convex.

Proof. It suffices to show that if x1 ∈ A1 and x2 ∈ A2 , then [x1 , x2 ] ⊆ A1 ∪A2 .
Since x2 ∈ A2 , Lemma 14.3.1 shows that [x1 , x2 ] intersects the point π2 (x1 ). By

Lemma 14.3.2, π2 (x1 ) ∈ A1 ∩ A2 . But then the two subsegments [x1 , π2 (x1 )] and
[π2 (x1 ), x2 ] are contained in A1 ∪ A2 , so the entire geodesic [x1 , x2 ] is contained in
A1 ∪ A2 .



14.4. Constructing R-trees by the stapling method
We now describe the “stapling method” for constructing R-trees. The following
definition is phrased for arbitrary metric spaces.
Definition 14.4.1. Let (V, E) be an unweighted undirected graph, let (Xv )v∈V
be a collection of metric spaces, and for each (v, w) ∈ E fix a set A(v, w) ⊆ Xv

14.4. CONSTRUCTING R-TREES BY THE STAPLING METHOD

241

−1
and an isometry ψv,w : A(v, w) → A(w, v) such that ψw,v = ψv,w
. Let ∼ be the
`
equivalence relation on v∈V Xv defined by the relations

x ∼ ψv,w (x) ∀(v, w) ∈ E ∀x ∈ A(v, w).

Then the stapled union of of the collection (Xv )v∈V with respect to the sets
(A(v, w))(v,w)∈E and the bijections (ψv,w )(v,w)∈E is the set
X=

st
a

Xv :=

v∈V

equipped with the path metric

(14.4.1)


d hv, xi, hw, yi = inf
















n
X



i=0












a

v∈V

Xv / ∼,

dvi (xi , yi )


v0 , . . . , vn ∈ V 





(vi , vi+1 ) ∈ E ∀i < n 





v0 = v, vn = w 

yi ∈ A(vi , vi+1 ) ∀i < n 






xi+1 = ψvi ,vi+1 (yi ) ∀i < n 




x0 = x, yn = y 

.

Note that d is finite as long as the graph (V, E) is connected. We leave it to the
reader to verify that in this case, d is a metric on X.
Example 14.4.2. If for each (v, w) ∈ E we fix a point p(v, w) ∈ Xv , then we
can let A(v, w) = {p(v, w)} and let ψv,w be the unique bijection between {p(v, w)}
and {p(w, v)}.

`st
Intuitively, the stapled union v∈V Xv is the metric space that results from
starting with the spaces (Xv )v∈V and for each (v, w) ∈ E, stapling the set A(v, w) ⊆

Xv with the set A(w, v) ⊆ Xw along the bijection ψv,w .

Definition 14.4.3 (Cf. Figure 14.4.1). We say that the consistency condition
is satisfied if for every 3-cycle u, v, w ∈ V , we have
(I) A(u, v) ∩ A(u, w) 6= , and

(II) for all z ∈ A(u, v) ∩ A(u, w), we have
(a) ψu,w (z) ∈ A(w, v) and
(b) ψw,v ψu,w (z) = ψu,v (z).

Obviously, the consistency condition is satisfied whenever (V, E) has no cycles.
Theorem 14.5.5 and Examples 14.5.1-14.5.10 below show how it can be satisfied in
many reasonable circumstances. Now we prove the main theorem of this chapter:
for a connected graph with contractible cycles, the consistency condition implies
that the stapled union of R-trees is an R-tree, if the staples are taken along convex
sets. More precisely:

242

14. R-TREES AND THEIR ISOMETRY GROUPS

Figure 14.4.1. In this diagram, the arrows represent the bijections ψvi ,vj , while the ovals represent the sets A(vi , vj ). The consistency condition (Definition 14.4.3) states that (I) each of the
shaded regions is nonempty, (IIa) shaded regions go to shaded regions, and (IIb) if you start in a shaded region and traverse the
diagram, then you will get back to where you started.

Theorem 14.4.4. Let (V, E) be a connected graph with contractible cycles, let
(Xv )v∈V be a collection of R-trees, and for each (v, w) ∈ E let A(v, w) ⊆ Xv be a
nonempty closed convex set and let ψv,w : A(v, w) → A(w, v) be an isometry such
−1
that ψw,v = ψv,w
. Assume that the consistency condition is satisfied. Then
`st
(i) The stapled union X = v∈V Xv is an R-tree.
(ii) The infimum in (14.4.1) is achieved when

(a) (vi )n0 = [v, w], and
(b) for each i < n, yi is the image of xi under the nearest-neighbor
projection to A(vi , vi+1 ).

Proof. We prove part (ii) first. For each (v, w) ∈ E, let πv,w : Xv → A(v, w)

be the nearest-neighbor projection; then πv,w is 1-Lipschitz. Now fix v ∈ V arbitrary. We define a map πv : X → Xv as follows. Fix x = hw, xi ∈ X, so that
x ∈ Xw . Let (vi )n0 = [v, w], and let

πv (x) = πv (w, x) = ψv1 ,v0 πv1 ,v0 · · · ψvn ,vn−1 πvn ,vn−1 (x).
Claim 14.4.5. The map πv is well-defined.

14.4. CONSTRUCTING R-TREES BY THE STAPLING METHOD

243

Proof. Fix (u, w) ∈ E and x ∈ A(u, w) and let y = ψu,w (x); we need to

show that πv (u, x) = πv (w, y). If w ∈ [v, u] or u ∈ [v, w] then the equality is
trivial, so by Corollary 14.2.7 we are reduced to proving the case where there exists
v ′ ∈ V such that (v ′ , w), (v ′ , u) ∈ E and v ′ ∈ [v, w], [v, u]. We have πv (u, x) =
πv (v ′ , ψu,v′ πu,v′ (x)) and πv (w, y) = πv (v ′ , ψw,v′ πw,v′ (y)), so to complete the proof
it suffices to show that
ψu,v′ πu,v′ (x) = ψw,v′ πw,v′ (y).

(14.4.2)

Since u, v ′ , w form a 3-cycle, part (I) of the consistency condition gives A(u, v ′ ) ∩
A(u, w) 6= . By Lemma 14.3.2, we have x′ := πu,v′ (x) ∈ A(u, v ′ )∩A(u, w). Apply-

ing part (IIa) of the consistency condition gives y ′′ := ψu,w (x′ ) ∈ A(w, v ′ ) and thus
d(x, A(u, v ′ )) = d(x, x′ ) = d(y, y ′′ ) ≤ d(y, A(w, v ′ )). A symmetric argument gives
d(y, A(w, v ′ )) ≤ d(x, A(u, v ′ )), so we have equality and thus y ′′ = y ′ := πw,v′ (y).
Applying part (IIb) of the consistency condition gives ψu,v′ (x′ ) = ψw,v′ (y ′ ), i.e.
(14.4.2) holds.

⊳

Since for each w ∈ V the map Xw ∋ x 7→ πv (w, x) ∈ Xv is 1-Lipschitz, the
map πv : X → Xv is also 1-Lipschitz.
Fix x = hv, xi, y = hw, yi ∈ X. Let (vi )n0 , (xi )n0 , and (yi )n0 be as in (ii), i.e.
(vi )n0 = [v, w], where x0 = x,
yi = πvi ,vi+1 (xi ) ∀i < n,
xi+1 = ψvi ,vi+1 (yi ) ∀i < n, and
yn = y.
We define a function f : X → Rn+1 as follows: for each z ∈ X, we let
n
f (z) = dvi (xi , πvi (z)) i=0 .

Then f is 1-Lipschitz, when Rn+1 is interpreted as having the max norm.
Claim 14.4.6. Fix z ∈ X and i = 0, . . . , n − 1. If fi+1 (z) > 0, then fi (z) ≥

ri := dvi (xi , yi ).

Proof. By contradiction, suppose that fi+1 (z) > 0 but fi (z) < dvi (xi , yi ).
Then zi+1 := πvi+1 (z) 6= xi+1 , but zi := πvi (z) ∈ B(xi , ri ) \ {yi }. In particular,

πvi+1 ,vi (zi ) = yi , so
(14.4.3)

zi+1 6= ψvi ,vi+1 πvi ,vi+1 (zi ).

On the other hand, since zi ∈
/ A(vi , vi+1 ), we have
(14.4.4)

zi 6= ψvi+1 ,vi πvi+1 ,vi (zi+1 ).

244

14. R-TREES AND THEIR ISOMETRY GROUPS

Write z = hw, zi. Then the definition of the maps (πv )v∈V together with (14.4.3),

(14.4.4) implies that vi ∈
/ [w, vi+1 ] and vi+1 ∈
/ [w, vi ]. Thus by Corollary 14.2.7,
there exists w′ ∈ V such that (w′ , vi ), (w′ , vi+1 ) ∈ E and w′ ∈ [w, vi ], [w, vi+1 ].

Let z ′ = πw′ (z), so that ψw′ ,vi πw′ ,vi (z ′ ) = zi and ψw′ ,vi+1 πw′ ,vi+1 (z ′ ) = zi+1 . Let
F = ψvi ,w′ (B(xi , ri ) ∩ A(vi , w′ )), and let πF : Xw′ → F be the nearest-neighbor
projection map. By Lemma 14.3.2, either πF (z ′ ) ∈ A(w′ , vi+1 ) or πw′ ,vi+1 (z ′ ) ∈ F .

Case 1: πF (z ′ ) ∈ A(w′ , vi+1 ). Since F ⊆ A(w′ , vi ) and πw′ ,vi (z ′ ) ∈ F , we have
πw′ ,vi (z ′ ) = πF (z ′ ) ∈ A(w′ , vi+1 ) and then part (IIa) of the consistency
condition gives zi = ψw′ ,vi πw′ ,vi (z ′ ) ∈ A(vi , vi+1 ), a contradiction.

Case 2: πw′ ,vi+1 (z ′ ) ∈ F . Since F ⊆ A(w′ , vi ), part (IIa) of the consistency condition gives zi+1 = ψw′ ,vi+1 πw′ ,vi+1 (z ′ ) ∈ A(vi+1 , vi ) and ψvi+1 ,vi (zi+1 ) ∈
ψw′ ,vi (F ) ⊆ B(xi , ri ). But then ψvi+1 ,vi (zi+1 ) = yi and thus zi+1 = xi+1 ,
a contradiction.

⊳
Thus f (X) is contained in the set
S = {(ti )n0 : ∀i = 0, . . . , n − 1 ti+1 > 0 ⇒ ti ≥ ri } ⊆ Rn+1 .
Now the function h : S → R defined by

h (ti )n0 = max [r0 + . . . + ri−1 + ti ]
i∈{0,...,n}
ti >0 if i>0

is Lipschitz 1-continuous with respect to the path metric of the max norm. Thus
since X is a path-metric space, h ◦ f : X → R is Lipschitz 1-continuous. Thus
d(x, y) ≥ h ◦ f (y) − h ◦ f (x) ≥ r0 + . . . + rn =

n
X

dvi (xi , yi ),

i=0

completing the proof of (ii).
For each x = hv, xi, y = hw, yi ∈ X, let
[x, y] = [x0 , y0 ]v0 ∗ · · · ∗ [xn , yn ]vn ,
where ∗ denotes the concatenation of geodesics, and (vi )n0 , (xi )n0 , and (yi )n0 are

as in (ii). Here [x, y]v denotes the image of the geodesic [x, y] under the map
Xv ∋ z → hv, zi ∈ X. Then by (ii), [x, y] is a geodesic connecting x and y. Thus
we have a family of geodesics ([x, y])x,y∈X .
We now prove that X is an R-tree, using the criteria of Lemma 3.1.12. Con-

dition (BII) is readily verified. So to complete the proof, we must demonstrate
(BIII). Fix x1 , x2 , x3 ∈ X distinct, and we show that two of the geodesics [xi , xj ]
have a nontrivial intersection. Write xi = hvi , xi i. If there is more than one possible
P
choice, choose (vi )31 so as to minimize i6=j dE (vi , vj ).

14.5. EXAMPLES OF R-TREES CONSTRUCTED USING THE STAPLING METHOD

245

Let w1 , w2 , w3 ∈ V be as in Lemma 14.2.6, with the convention that w1 = w2 =

w3 = w if we are in Case 1 of Lemma 14.2.6.

Case A: For some i, vi 6= wi . Choose j, k such that i, j, k are distinct. Then there
exists a vertex w ∈ V adjacent to vi such that w ∈ [vi , vj ] ∩ [vi , vk ]. The
choice of (vi )31 guarantees that xi ∈
/ A(vi , w), so that [xi , πvi ,w (xi )]vi forms

a common initial segment of the geodesics [xi , xj ] and [xi , xk ].
Case B: For all i, vi = wi . Then either v1 = v2 = v3 , or v1 , v2 , v3 form a cycle.

Case B1: Suppose that v1 = v2 = v3 = v. Then since Xv is an R-tree, there
exist distinct i, j, k ∈ {1, 2, 3} such that the geodesics [xi , xj ]v and

[xi , xk ]v have a common initial segment.
Case B2: Suppose that v1 , v2 , v3 form a cycle. Then by part (I) of the consistency condition A(v1 , v2 ) ∩ A(v1 , v3 ) 6= , so by Lemma 14.3.3 the
set F = A(v1 , v2 ) ∪ A(v1 , v3 ) is convex. But the choice of (vi )31 guar-

antees that x1 ∈
/ F , so that [x1 , πF (x1 )]v1 forms a common initial
segment of the geodesics [x1 , x2 ] and [x1 , x3 ].


14.5. Examples of R-trees constructed using the stapling method
We give three examples of ways to construct R-trees using the stapling method
so that the resulting R-tree admits a natural isometric action.
Example 14.5.1 (Cone construction again). Let (Z, D) be a complete ultrametric space, let V = Z and E = V × V \ {(v, v) : v ∈ V }, and for each v ∈ V

let Xv = R. For each v, w ∈ V let A(v, w) = [log D(v, w), ∞), and let ψv,w be the
identity map. Since (V, E) is a complete graph, it is connected and has contractible

cycles. Part (IIa) of the consistency condition is equivalent to the ultrametric inequality for D, while parts (I) and (IIb) are obvious. Thus we can consider the
`st
stapled union X = v∈V Xv . One can verify that the stapled union is isomet-

ric to the R-tree X considered in the proof of Theorem 14.1.1. Indeed, the map
hz, ti 7→ hz, et i provides the desired isometry. Note that the map ι constructed in

Theorem 14.1.1 can be described in terms of the stapled union as follows: For each
z ∈ Z, ι(z) is the image of −∞ under the isometric embedding of Xz ≡ R into X.

(The image of +∞ is ∞).

Our next example is a type of Schottky product which we call a “pure Schottky
product”. To describe it, it will be convenient to introduce the following terminology:
Definition 14.5.2. If Γ is a group, a function k · k : Γ → [0, ∞) is called tree-

geometric if there exist an R-tree X, a distinguished point o ∈ X, and an isometric

246

14. R-TREES AND THEIR ISOMETRY GROUPS

action φ : Γ → Isom(X) such that
kφ(γ)k = kγk ∀γ ∈ Γ.
Example 14.5.3. Theorem 14.1.5 gives a sufficient but not necessary condition
for a function to be tree-geometric.
Remark 14.5.4. If the group Γ is countable, then whenever Γ is a treegeometric function, the R-tree X can be chosen to be separable.
Proof. Without loss of generality, we may replace X by the convex hull of
Γ(o).

Theorem 14.5.5 (Cf. Figure 14.5.1). Let (Hj )j∈J be a (possibly infinite) collection of groups and for each j ∈ J let k · k : Hj → [0, ∞) be a tree-geometric
function. Then the function k · k : G = ∗j∈J Hj → [0, ∞) defined by

(14.5.1)

kh1 · · · hn k := kh1 k + · · · + khn k

(assuming h1 . . . hn is given in reduced form) is a tree-geometric function.
Proof. For each j ∈ J write Hj ≤ Isom(Xj ) and khk = d(oj , h(oj )) ∀h ∈ Hj
for some R7-tree Xj and for some distinguished point oj ∈ Xj . Let V = J × G,

and for each (j, g) ∈ V let Xv = Xj . Let

E1 = {((j, g), (k, g)) : j 6= k, g ∈ G}
E2 = {((j, g), (j, gh)) : j ∈ J, g ∈ G, h ∈ Hj \ {e}}
E = E1 ∪ E2 .
Claim 14.5.6. Any cycle in (V, E) is contained in a complete graph of one of
the following forms:
(14.5.2)

{(j, gh) : h ∈ Hj } (j ∈ J, g ∈ G fixed),

(14.5.3)

{(j, g) : j ∈ J} (g ∈ G fixed).

In particular, (V, E) is a graph with contractible cycles.
Proof. Let (vi )n0 be a cycle in V , and for each i = 0, . . . , n − 1 let ei =
(vi , vi+1 ). By contradiction suppose that (vi )n0 is not contained in a complete graph
of one of the forms (14.5.2),(14.5.3). Without loss of generality suppose that (vi )n0
is minimal with this property. Then no two consecutive edges ei , ei+1 can lie in the
same set Ek . After reindexing if necessary, we find ourselves in the position that
ei ∈ E2 for i even and ei ∈ E1 for i odd. Write v0 = (j1 , g); then
v0 = (j1 , g), v1 = (j1 , gh1 ), v2 = (j2 , gh1 ), v3 = (j2 , gh1 h2 ), [etc.]

14.5. EXAMPLES OF R-TREES CONSTRUCTED USING THE STAPLING METHOD

247

with hi ∈ Hji , ji 6= ji+1 . Since G is a free product, this contradicts that vn = v0 .

⊳

For each (v, w) = ((j, g), (k, g)) ∈ E1 , we let A(v, w) = {oj } and we let

ψv,w (oj ) = oj . For each (v, w) = ((j, g), (j, gh)) ∈ E2 , we let A(v, w) = Xj and
we let ψv,w = h−1 . Claim 14.5.6 then implies the consistency condition. Consider
`
`
`st
the stapled union X = (j,g)∈V Xj = (j,g)∈V Xj / ∼. Elements of (j,g)∈V Xj
consist of pairs ((j, g), x), where g ∈ G and x ∈ Xj . We will abuse notation by
writing ((j, g), x) = (j, g, x) and h(j, g), xi = hj, g, xi. Then the “staples” are given
by the relations
(j, g, oj ) ∼ (k, g, ok ) [g ∈ G, j, k ∈ J],
(j, gh, x) ∼ (j, g, h(x)) [g ∈ G, j ∈ J, h ∈ Hj , x ∈ Xj ].
`
Now consider the following action of G on (j,g)∈V Xj :

g1 (j, g2 , x) = (j, g1 g2 , x).

Since the “staples” are preserved by this action, it descends to an action on the
stapled union X. To finish the proof, we need to show that d(o, g(o)) = kgk ∀g ∈ G,

where o = hj, e, oj i ∀j ∈ J, and k · k is given by (14.5.1). Indeed, fix g ∈ G and

write g = h1 · · · hn , where for each i = 1, . . . , n, hi ∈ Hji \ {e} for some j ∈ J, and
ji 6= ji+1 ∀i. For each i = 0, . . . , n let gi = h1 · · · hi , and for each i = 1, . . . , n let
(1)

vi
(1)

(2)

(2)

= (ji , gi−1 ), vi
(1)

(1)

= (ji , gi ).

(2)

Then the sequence (v1 , v1 , v2 , . . . , vn , vn ) is a geodesic whose endpoints are
(k)

(k)

(j1 , e) and (jn , g). We compute the sequences (xi ), (yi ) as in Theorem 14.4.4(ii):
(1)

xi

(1)

= oji , yi

(2)

= oji , xi

(2)

= h−1
i (oji ), yi

= oji ,

It follows that
d(o, g(o)) =

n X
2
X
i=1 j=1

(j)

(j)

d(xi , yi ) =

n
X
i=1

khi k = kgk,

which completes the proof.



Definition 14.5.7. Let (Hj )j∈J and G be as in Theorem 14.5.5. If we write
G ≤ Isom(X) and kgk = d(o, g(o)) ∀g ∈ G for some R-tree X and some distinguished point o ∈ X, then we call (X, G) the pure Schottky product of (Hj )j∈J . (It
is readily verified that every pure Schottky product is a Schottky product.)
Proposition 14.5.8. The Poincaré set of a pure Schottky product H1 ∗ H2 can

be computed by the formula

s ∈ ∆(H1 ∗ H2 ) ⇔ (Σs (H1 ) − 1)(Σs (H2 ) − 1) ≥ 1.

248

14. R-TREES AND THEIR ISOMETRY GROUPS

Figure 14.5.1. The Cayley graph of F2 (Z), interpreted as the
pure Schottky product H1 ∗ H2 , where H1 = H2 = Z is interpreted as acting on X1 = X2 = R by translation. The horizontal
lines correspond to copies of R which correspond to vertices of the
form (1, g), while the vertical lines correspond to copies of R which
correspond to vertices of the form (2, g). The intersection points
between horizontal and vertical lines are the staples which hold the
tree together.

Proof. Let
E = (H1 \ {id})(H2 \ {id}),
so that
G=

[

H2 E n H1 .

n≥0

Then by (14.5.1), we have for all s ≥ 0
Σs (G) =

X

e

−skgk

=

∞
X
X

X

X

n=0 h0 ∈H2 g1 ,...,gn ∈E hn+1 ∈H1

g∈G

= Σs (H2 )Σs (H1 )

∞
X

n=0

= Σs (H2 )Σs (H1 )

∞
X

n=0

This completes the proof.




X

g∈E

e−s[kh0 k+

Pn
1

kgi k+khn+1 k]

n

e−skgk 

n
(Σs (H1 ) − 1)(Σs (H2 ) − 1) .



Proposition 14.5.8 generalizes to the case of more than two groups as follows:
Proposition 14.5.9. The Poincaré set of a finite pure Schottky product
G = ∗kj=1 Hj

14.5. EXAMPLES OF R-TREES CONSTRUCTED USING THE STAPLING METHOD

249

can be computed by the formula
s ∈ ∆(H1 ∗ H2 ) ⇔ ρ(As ) ≥ 1,
where ρ denotes spectral radius, and As denotes the matrix whose (j, j ′ )th entry is

Σ (H ) j ′ 6= j
s
j
.
(As )j,j ′ =
0
j′ = j
Proof. Let J = {1, . . . , k}. Then
G=

∞
[

[

n=0 j1 ,...,jn ∈J
j1 6=···6=jn

{h1 · · · hn : h1 ∈ Hj1 , · · · , hn ∈ Hjn }.

So by (14.5.1), we have for all s ≥ 0
Σs (G) =

X

e−skgk =

∞
X

X

∞
X

X

X

n=0 j1 ,...,jn ∈J h1 ∈Hj1
j1 6=···6=jn

g∈G

=

n
Y

n=0 j1 ,...,jn ∈J i=1
j1 6=···6=jn

···

X

e−s

Pn

khi k

hn ∈Hjn

(Σs (Hji ) − 1)


Σs (H1 ) − 1


..

=1+
[1 · · · 1]Asn−1 
.


n=1
Σs (Hn ) − 1

= ∞ ρ(A ) ≥ 1
s
.
< ∞ ρ(A ) < 1
∞
X

1



s

This completes the proof.



Note that only the last step (the series converges or diverges according to
whether or not the spectral radius is at least one) uses the hypothesis that J is
finite.
Our last example of an R-tree constructed using the stapling method is similar
to the method of pure Schottky products, but differs in important ways:
Example 14.5.10 (Geometric products). Let Y be an R-tree, let P ⊆ Y be

a set, and let (Γp )p∈P be a collection of abstract groups. Let Γ = ∗p∈P Γp . Let
V = Γ, and let
E = {(γ, γα) : γ ∈ Γ, α ∈ Γp \ {e}}.
For each v ∈ V , let Xv = Y . For each (v, w) = (γ, γα) ∈ E, where γ ∈ Γ and
α ∈ Γp \ {e}, we let A(v, w) = {p}, and we let ψv,w (p) = p. In a manner similar

to the proof of Claim 14.5.6, one can check that every cycle in (V, E) is contained

250

14. R-TREES AND THEIR ISOMETRY GROUPS

in one of the complete graphs γΓp ⊆ V (γ ∈ Γ, p ∈ P ), so (V, E) has contractible

cycles. The consistency condition is trivial. Thus we can consider the stapled union
`st
X = v∈V Xv , which admits a natural left action φ : Γ → Isom(X):
ι(γ)(hv, xi) = hγv, xi.

We let G = φ(Γ), and we call the pair (X, G) the geometric product of Y with
(Γp )p∈P .

Note that if (X, G) is the geometric product of Y with (Γy )y∈A , then for all
g = (p1 , γ1 ) · · · (pn , γn ) ∈ G, we have
(14.5.4)

kgk = d(o, p1 ) +

n−1
X

d(pi , pi+1 ) + d(pn , o).

i=1

To compare this formula with (14.5.1), we observe that if n = 1, then we get
k(a, γ)k = 2d(o, a), so that
k(p1 , γ1 )k + · · · + k(pn , γn )k =

n
X

2d(o, pi )

i=1

= d(o, p1 ) +

n−1
X

[d(o, pi ) + d(o, pi+1 )] + d(o, pn ).

i=1

So if (X, G) is a geometric product, then the right hand side of (14.5.1) exceeds
Pn−1
the left hand side by i=1
2hpi |pi+1 io . The formula (14.5.4) is more complicated

to deal with because its terms depend on the relation between the neighborhing
points pi and pi+1 , rather than just on the individual terms pi . In particular, it
is more difficult to compute the Poincaré exponent of a geometric product than it
is to compute the Poincaré exponent of a group coming from Theorem 14.5.5. We
will investigate the issue of computing Poincaré exponents of geometric products
in [57], as well as other topics related to the geometry of these groups.

Example 14.5.11 (Cf. Figure 14.5.3). Let (an )∞
1 be an increasing sequence of
nonnegative real numbers, and let (bn )∞
1 be a sequence of nonnegative real numbers.
Let
∞
[
Y = ([0, ∞) × {0}) ∪
({an } × [0, bn ])
n=1

with the path metric induced from R2 . Let P = {pn : n ∈ N}, where pn = (an , bn ).
Then

(14.5.5)

d(pn , pm ) = bn + bm + |an − am | ∀m 6= n,

14.5. EXAMPLES OF R-TREES CONSTRUCTED USING THE STAPLING METHOD

251

(Y, bab)
(Y, ba)
(Y, b)
0

1

(Y, e)
(Y, a)
(Y, ab)
(Y, aba)

Figure 14.5.2. The geometric product of Y with (Γp )p∈P , where
Y = [0, 1], P = {0, 1}, Γ0 = {e, γ0 } ≡ Z2 , and Γ1 = {e, γ1 } ≡ Z2 .
In the left hand picture, copies of Y are drawn as horizontal lines
and identifications between points in different copies are drawn
as vertical lines. The right hand picture is the result of stapling
together certain pairs of points in the left hand picture.
so (14.5.4) would become
kgk = b1 + a1 +
=

n
X
i=1

n−1
X
i=1

[bi + bi+1 + |ai+1 − ai |] + bn + an

2bi + a1 +

n−1
X
i=1

|ai+1 − ai | + an .

This formula exhibits clearly the fact that the relation between neighborhing points
pi and pi+1 is involved, via the appearance of the term |ai+1 − ai |.
Proposition 14.5.12. Let (X, G) be the geometric product of Y with (Γp )p∈P ,
where P ⊆ Y .
(i) If
(14.5.6)

inf{d(y, z) : y, z ∈ E, y 6= z} > 0,
then G = hGa ia∈E is a global weakly separated Schottky product. If furthermore

(14.5.7)

inf{D(y, z) : y, z ∈ E, y 6= z} > 0,
then G is strongly separated.

(ii) X is proper if and only if all three of the following hold: Y is proper,
#(Γa ) < ∞ for all a ∈ E, and #(E ∩ B(o, ρ)) < ∞ for all ρ > 0.
Proof of (i). Suppose that (14.5.6) holds, and for each p ∈ P , let
Up = {hg1 · · · gn , yi ∈ X : g1 ∈ Gp } ∪ {hid, yi : y ∈ B(p, ε)},

252

14. R-TREES AND THEIR ISOMETRY GROUPS

0

∞

Figure 14.5.3. The set Y of Example 14.5.11. The points at the
tops of the vertical lines are “branch points” which correspond to
fixed points in the geometric product (X, G). If a geodesic in the
geometric product is projected down to Y , the result will be a
sequence of geodesics, each of which starts and ends at one of the
indicated points (either o, an element of P , or ∞).
where ε ≤ inf{d(y, z) : y, z ∈ P, y 6= z}/2. Then (Up )p∈P a global Schottky system
for G. If (14.5.7) also holds, then it is strongly separated, because

inf{D(Up , Uq ) : p 6= q} ≥ inf{D(y, z) : y, z ∈ P, y 6= z} − 2ε
can be made positive if ε is sufficiently small. Finally, if we go back to assuming
only that (14.5.6) holds, then (Up )p∈P is still weakly separated, because (14.5.7)
holds for finite subsets.



Proof of (ii). The necessity of these conditions is obvious; conversely, suppose they hold. Fix ρ > 0 and x = hg, yi ∈ BX (o, ρ); by (14.4.1), we have
d(o, p1 ) + d(p1 , p2 ) + . . . + d(pn−1 , pn ) + d(pn , y) ≤ ρ,
where g = h1 · · · hn , hi ∈ Gpi \ {id}, pi ∈ P , and pi 6= pi+1 for all i. It follows
that kpi k ≤ ρ for all i = 1, . . . , n, i.e. pi ∈ P ∩ B(o, ρ). In particular, letting

ε = minp,q∈P ∩B(o,ρ) d(a, b), we have (n − 1)ε ≤ ρ, or equivalently n ≤ 1 + ρ/ε. It
follows that
[
[
(Gp1 \ {id}) · · · (Gpn \ {id}),
g∈
n≤1+ρ/ε p1 ,...,pn ∈P ∩B(o,ρ)

a finite set. Thus, BX (o, ρ) is contained in the union of finitely many compact sets
of the form BY (o, ρ) × {g} ⊆ X, and is therefore compact.



Part 4

Patterson–Sullivan theory

This part will be divided as follows: In Chapter 15 we recall the definition of
quasiconformal measures, and we prove basic existence and non-existence results.
In Chapter 16, we prove Theorem 1.4.1 (Patterson–Sullivan theorem for groups
of divergence type). In Chapter 17, we investigate the geometry of quasiconformal
measures of geometrically finite groups, and we prove a generalization of the Global
Measure Formula (Theorem 17.2.2) as well as giving various necessary and/or sufficient conditions for the Patterson–Sullivan measure of a geometrically finite group
to be doubling (§17.4) or exact dimensional (§17.5).

CHAPTER 15

Conformal and quasiconformal measures
Standing Assumption. Throughout the final part of the monograph, i.e. in
Chapters 15-17, we fix (X, d, o, b) as in §4.1, and a group G ≤ Isom(X).
15.1. The definition
Conformal measures, introduced by S. G. Patterson [142] and D. P. Sullivan
[161], are an important tool in studying the geometry of the limit set of a Kleinian
group. Their definition can be generalized directly to the case of a group acting
on a strongly hyperbolic metric space, but for a hyperbolic metric space which is
not strongly hyperbolic, a multiplicative error term is required. Thus we make the
following definition (cf. [53, Definition 4.1]):
Definition 15.1.1. For each s ≥ 0, a nonzero measure1 µ on ∂X is called
s-quasiconformal 2 if
Z
(15.1.1)
µ(g(A)) ≍×
[g ′ (ξ)]s dµ(ξ)
A

for every g ∈ G and for every Borel set A ⊆ ∂X. If X is strongly hyperbolic and if

equality holds in (15.1.1), then µ is called s-conformal.

Remark 15.1.2. For two measures µ1 , µ2 , write µ1 ≍× µ2 if µ1 and µ2 are in
the same measure class and if the Radon–Nikodym derivative dµ1 /dµ2 is bounded
from above and below. Then a measure µ is s-quasiconformal if and only if
µ ◦ g ≍× [g ′ (ξ)]s µ,
and is s-conformal if X is strongly hyperbolic and if equality holds.
Remark 15.1.3. One might ask whether it is possible to generalize the notions
of conformal and quasiconformal measures to semigroups. However, this appears
to be difficult. The issue is that the condition (15.1.1) is sometimes impossible to
satisfy for measures supported on Λ – for example, it may happen that there exist
g1 , g2 ∈ G such that g1 (Λ) ∩ g2 (Λ) = , in which case letting A = ∂X \ Λ in (15.1.1)
shows both that Supp(µ) ⊆ g1 (Λ) and that Supp(µ) ⊆ g2 (Λ), and thus that µ = 0.
1In this monograph, “measure” always means “nonnegative finite Borel measure”.
2Not to be confused with the concept of a quasiconformal map, cf. [92].
255

256

15. CONFORMAL AND QUASICONFORMAL MEASURES

One may try to fix this by changing the formula (15.1.1) somehow, but it is not
clear what the details of this should be.
15.2. Conformal measures
Before discussing quasiconformal measures, let us consider the relation between
conformal measures and quasiconformal measures. Obviously, every conformal measure is quasiconformal. In the converse direction we have:
Proposition 15.2.1. Suppose that G is countable and that X is strongly hyperbolic. Then for every s ≥ 0, if µ is an s-quasiconformal measure, then there

exists an s-conformal measure ν satisfying ν ≍× µ.

Proof. For each g ∈ G, let fg : ∂X → (0, ∞) be a Radon–Nikodym derivative
of µ ◦ g with respect to µ. Since µ is s-quasiconformal, we have for µ-a.e. ξ ∈ ∂X
(15.2.1)

fg (ξ) ≍× [g ′ (ξ)]s .

Since G is countable, the set of ξ ∈ ∂X for which (15.2.1) holds for all g ∈ G is of
full µ-measure. In particular, if
fg (ξ)
,
′
s
g∈G [g (ξ)]

f (ξ) = sup

then f (ξ) ≍× 1 for µ-a.e. ξ ∈ X. Now for each g, h ∈ G, the equality µ ◦ (gh) =
(µ ◦ g) ◦ h implies that
fgh (ξ) = fg (h(ξ))fh (ξ) for µ-a.e. ξ ∈ ∂X.
Combining with the chain rule for metric derivatives, we have
fg (h(ξ)) fh (ξ)
fgh (ξ)
= ′
for µ-a.e. ξ ∈ ∂X.
[(gh)′ (ξ)]s
[g (h(ξ))]s [h′ (ξ)]s
Note that we are using the strong hyperbolicity assumption here to get equality
rather than a coarse asymptotic. Taking the supremum over all g gives
f (ξ) = f (h(ξ))

fh (ξ)
for µ-a.e. ξ ∈ ∂X.
[h′ (ξ)]s

We now claim that ν := f µ is an s-conformal measure. Indeed,
f (g(ξ)) dµ ◦ g
f (g(ξ))
dν ◦ g
(ξ) =
(ξ) =
fg (ξ) = [g ′ (ξ)]s for µ-a.e. ξ ∈ ∂X.
dν
f (ξ)
dµ
f (ξ)

15.3. Ergodic decomposition
Let M(∂X) denote the set of all measures on ∂X, and let M1 (∂X) denote the

set of all probability measures on ∂X.

15.3. ERGODIC DECOMPOSITION

257

Definition 15.3.1. A measure µ ∈ M(∂X) is ergodic if for every G-invariant

Borel set A ⊆ ∂X, we have µ(A) = 0 or µ(∂X \ A) = 0.

It is often useful to be able to write a non-ergodic measure as the convex
combination of ergodic measures. To make this rigorous, suppose that X is complete
and separable, so that bord X and ∂X are Polish spaces. Then ∂X together with its
Borel σ-algebra forms a standard Borel space. Let B denote the smallest σ-algebra
on M(∂X) with the following property:

Property 15.3.2. For every bounded Borel-measurable function f : ∂X → R,
the function
Z
µ 7→ f dµ

is a B-measurable map from M(∂X) to R.

Then (M(∂X), B) is a standard Borel space. We may now state the following
theorem:
Proposition 15.3.3 (Ergodic decomposition of quasiconformal measures). We
suppose that G is countable and that X is separable. Fix s ≥ 0.

(i) For every s-quasiconformal measure µ, there is a measure µ
b on M1 (∂X)
which satisfies
Z
(15.3.1)
µ(A) = ν(A) db
µ(ν) for every Borel set A ⊆ ∂X
and gives full measure to the set of ergodic s-quasiconformal measures.3

(ii) If X is strongly hyperbolic, then for every s-conformal measure µ, there is
a unique measure µ
b on M(∂X) which satisfies (15.3.1) and which gives
full measure to the set of ergodic s-conformal measures.

Remark 15.3.4. Note that we have uniqueness in (ii) but not in (i).

Proof of Proposition 15.3.3. Both cases of the proposition are essentially
special cases of [82, Theorem 1.4], as we now demonstrate:
(i) Let µ be an s-quasiconformal measure. Let ̺ : G × ∂X → R satisfy [82,

(1.1)-(1.3)]. Then by [82, Theorem 1.4], there is a measure µ
b satisfying (15.3.1) supported on the set of ergodic probability measures which

are “̺-admissible” (in the terminology of [82]). But by [82, (1.1)], we
have b̺(g,ξ) ≍× g′ (ξ)s for µ-a.e. ξ ∈ ∂X, say for all ξ ∈ ∂X \ S, where

µ(S) = 0. Then every ̺-admissible measure ν satisfying ν(S) = 0 is squasiconformal. But by (15.3.1), ν(S) = 0 for µ
b-a.e. ν, so µ
b-a.e. ν is
s-quasiconformal.

3If A is a non-measurable set, then a measure µ gives full measure to A if and only if A contains a

measurable set of full µ-measure. Thus we do not need to check whether or not the set of ergodic
s-quasiconformal measures is a measurable set in M1 (∂X).

258

15. CONFORMAL AND QUASICONFORMAL MEASURES

(ii) Let µ be an s-conformal measure. Let ̺ : G × ∂X → R satisfy [82, (1.1)-

(1.3)]. Then by [82, (1.1)], we have b̺(g,ξ) = g ′ (ξ)s for µ-a.e. ξ ∈ ∂X, say
for all ξ ∈ ∂X \ S, where µ(S) = 0. Then for every measure ν satisfying
ν(S) = 0, ν is ̺-admissible if and only if ν is s-conformal. By [82,
Theorem 1.4], there is a unique measure µ
b satisfying (15.3.1) supported
on the set of ̺-admissible ergodic probability measures; such a measure is
also unique with respect to satisfying (15.3.1) being supported on the set

of s-conformal ergodic measures.

Corollary 15.3.5. Suppose that G is countable and that X is separable, and
fix s ≥ 0. If there is an s-(quasi)conformal measure, then there is an ergodic s(quasi)conformal measure.
In the sequel, we will be concerned with when an s-quasiconformal measure is
unique up to coarse asymptotic. This is closely connected with ergodicity:
Proposition 15.3.6. Suppose that G is countable and that X is separable, and
fix s ≥ 0. Suppose that there is an s-quasiconformal measure µ. The following are
equivalent:

(A) µ is unique up to coarse asymptotic i.e. µ ≍× µ
e for any s-quasiconformal
measure µ
e.
(B) Every s-quasiconformal measure is ergodic.

If in addition X is strongly hyperbolic, then (A)-(B) are equivalent to
(C) There is exactly one s-conformal probability measure.

Proof of (A) ⇒ (B). If µ is a non-ergodic s-quasiconformal measure, then

there exists a G-invariant set A ⊆ ∂X such that µ(A), µ(∂X \ A) > 0. But then
ν1 = µ ↿ A and ν2 = µ ↿ ∂X \ A are non-asymptotic s-quasiconformal measures, a
contradiction.



Proof of (B) ⇒ (A). Suppose that µ1 , µ2 are two s-quasiconformal measures.

Then the measure µ = µ1 + µ2 is also s-quasiconformal, and therefore ergodic. Let
fi be a Radon–Nikodym derivative of µi with respect to µ. Then for all g ∈ G,
(15.3.2)

fi ◦ g(ξ) =

It follows that

[g ′ (ξ)]s dµi
dµi ◦ g
(ξ) ≍× ′
(ξ) = fi (ξ) for µ-a.e. ξ ∈ ∂X.
dµ ◦ g
[g (ξ)]s dµ

hi (ξ) := sup fi ◦ g(ξ) ≍× fi (ξ) for µ-a.e. ξ ∈ ∂X.
g∈G

15.4. QUASICONFORMAL MEASURES

259

But the functions hi are G-invariant, so since µ is ergodic, they are constant µ-a.e.,
say hi = ci . It follows that µi ≍× ci µ; since µi 6= 0, we have ci > 0 and thus
µ1 ≍ × µ2 .

Proof of (B) ⇒ (C). The existence of an s-conformal measure is guaranteed

by Proposition 15.2.1. If µ1 , µ2 are two s-conformal measures, then the Radon–

Nikodym derivatives fi = dµi /d(µ1 + µ2 ) satisfy (15.3.2) with equality, so fi = ci
for some constants ci . It follows that µ1 = (c1 /c2 )µ2 , and so if µ1 , µ2 are probability
measures then µ1 = µ2 .



Proof of (C) ⇒ (A). Follows immediately from Proposition 15.2.1.



15.4. Quasiconformal measures
We now turn to the deeper question of when a quasiconformal measure exists in
the first place. To approach this question we begin with a fundamental geometrical
lemma about quasiconformal measures:
Lemma 15.4.1 (Sullivan’s Shadow Lemma, cf. [161, Proposition 3], [152,
§1.1]). Fix s ≥ 0, and let µ be a s-quasiconformal measure on ∂X which is not
a pointmass. Then for all σ > 0 sufficiently large and for all g ∈ G,
µ(Shad(g(o), σ)) ≍×,σ,µ b−skgk .
Proof. We have
µ(Shad(g(o), σ)) ≍×,µ

Z

g′

g−1 (Shad(g(o),σ))

s

dµ

(by the definition of s-quasiconformality)
Z
s
g ′ dµ
=
Shadg−1 (o) (o,σ)

≍×,σ

Z

b−skgk dµ

Shadg−1 (o) (o,σ)

(by the Bounded Distortion Lemma 4.5.6)

=
b−skgk µ Shadg−1 (o) (o, σ) .

Thus, to complete the proof, it is enough to show that

µ Shadg−1 (o) (o, σ) ≍×,µ,σ 1,

assuming σ is sufficiently large (depending on µ). The upper bound is automatic since µ is finite. Now, since by assumption µ is not a pointmass, we have
#(Supp(µ)) ≥ 2. Choose distinct ξ1 , ξ2 ∈ Supp(µ), and let ε = D(ξ1 , ξ2 )/3. By the

260

15. CONFORMAL AND QUASICONFORMAL MEASURES

Big Shadows Lemma 4.5.7, we have
Diam(∂X \ Shadg−1 (o) (o, σ)) ≤ ε
for all σ > 0 sufficiently large (independent of g). Now since
D(B(ξ1 , ε), B(ξ2 , ε)) ≥ ε,
it follows that
∃i = 1, 2 B(ξi , ε) ⊆ Shadg−1 (o) (o, σ)
and thus

2


µ Shadg−1 (o) (o, σ) ≥ min µ B(ξi , ε) > 0.
i=1

The right hand side is independent of g, which completes the proof.



Sullivan’s Shadow Lemma suggests that in the theory of quasiconformal measures, there is a division between those measures which are pointmasses and those
which are not. Let us first consider the easier case of a pointmass quasiconformal measure, and then move on to the more interesting theory of non-pointmass
quasiconformal measures.
15.4.1. Pointmass quasiconformal measures.
Proposition 15.4.2. A pointmass δξ is s-quasiconformal if and only if
(I) ξ ∈ ∂X is a global fixed point of G, and
(II) either
(IIA) ξ is neutral with respect to every g ∈ G, or
(IIB) s = 0.
Proof. To begin we recall that g ′ (ξ) denotes the dynamical derivative, cf.
Proposition 4.2.12. For each ξ ∈ ∂X,
δξ is s-quasiconformal ⇔ δξ ◦ g ≍× (g ′ )s δξ ∀g ∈ G

⇔ g(ξ) = ξ and [g ′ (ξ)]s ≍× 1 ∀g ∈ G

⇔ g(ξ) = ξ and [g ′ (ξ)]s = 1 ∀g ∈ G

⇔ g(ξ) = ξ and (g ′ (ξ) = 1 or s = 0) ∀g ∈ G.


Corollary 15.4.3.
(i) If G is of general type, then no pointmass is s-quasiconformal for any
s ≥ 0.

(ii) If G is loxodromic, then no pointmass is s-quasiconformal for any s > 0.

15.4. QUASICONFORMAL MEASURES

261

15.4.2. Non-pointmass quasiconformal measures. Next we will ask the
following question: Given a group G, for what values of s does a non-pointmass
quasiconformal measure exist, and when is it unique up to coarse asymptotic?
We first recall the situation in the Standard Case, where the answers are wellknown. The first result is the Patterson–Sullivan theorem [161, Theorem 1], which
states that any discrete subgroup G ≤ Isom(Hd ) admits a δG -conformal measure
supported on Λ. It is unique up to a multiplicative constant if G is of divergence
type ([138, Theorem 8.3.5] together with Proposition 15.3.6). The next result is
negative, stating that if s < δG , then G admits no non-pointmass s-conformal
measure. From these results and from Corollary 15.4.3, it follows that if G is
of general type, then δG is the infimum of s for which there exists an s-conformal
measure [161, Corollary 4]. Finally, for s > δG , an s-conformal measure on Λ exists
if and only if G is not convex-cocompact ([10, Theorem 4.1] for ⇐, [138, Theorem
4.4.1] for ⇒); no nontrivial conditions are known which guarantee uniqueness in

this case.
We now generalize the above results to the setting of hyperbolic metric spaces,
replacing the Poincaré exponent δG with the modified Poincaré exponent δeG , and
the notion of divergence type with the notion of generalized divergence type. By

Proposition 8.2.4(ii), our theorems will reduce to the known results in the case of
a strongly discrete group.
We begin with the negative result, as its proof is the easiest:

Proposition 15.4.4 (cf. [161, p.178]). For any s < δeG , there does not exist a
non-pointmass s-quasiconformal measure.
Proof. By contradiction, suppose that µ is a non-pointmass s-quasiconformal
measure. Let σ > 0 be large enough so that Sullivan’s Shadow Lemma 15.4.1 holds,
and let τ > 0 be the implied constant of (4.5.2) from the Intersecting Shadows
Lemma 4.5.4. Let Sτ +1 be a maximal (τ + 1)-separated subset of G(o). Fix n ∈ N,

and let An be the nth annulus An = B(o, n) \ B(o, n − 1). Now by the Intersecting

Shadows Lemma 4.5.4, the shadows Shad(x, σ) x∈Sτ +1 ∩An are disjoint, and so by
Sullivan’s Shadow Lemma 15.4.1

1 ≍×,µ µ(∂X) ≥

X

µ(Shad(x, σ))

x∈Sτ +1 ∩An

≍×,σ,µ
≍× b

X

b−skxk

x∈Sτ +1 ∩An

−sn

#(Sτ +1 ∩ An ).

262

15. CONFORMAL AND QUASICONFORMAL MEASURES

Thus for all t > s,
Σt (Sτ +1 ) ≍×

X

n∈N

b−tn #(Sτ +1 ∩ An ) .×,σ,µ

X

n∈N

b(s−t)n < ∞.

But this implies that δeG ≤ t (cf. (8.2.2)); letting t ց s gives δeG ≤ s, contradicting
our hypothesis.

Remark 15.4.5. The above proof shows that if there exists a non-pointmass
e
δ-conformal measure, then
e

#(Sτ +1 ∩ An ) .× bδn ∀n ≥ 1.

In particular, if δe > 0 then summing over n = 1, . . . , N gives
e

#(Sτ +1 ∩ B(o, N )) .× bδN ∀n ≥ 1.

If G is strongly discrete, then for all ρ > 0,
NX,G (ρ) = #{g ∈ G : kgk ≤ ρ} .× #(Sτ +1 ∩ B(o, ρ + τ + 1))
.× bδ⌈ρ+τ +1⌉
≍× bδρ .
The bound NX,G (ρ) .× bδρ in fact holds without assuming the existence of a
δ-conformal measure; see Corollary 16.7.1.
Next we study hypotheses which guarantee the existence of a δeG -quasiconformal
measure. In particular, we will show that if δeG < ∞ and if G is of compact type
or of generalized divergence type, then there exists a δeG -quasiconformal measure.
The first case we consider now, while the case of a group of generalized divergence
type will be considered in Chapter 16.

Theorem 15.4.6 (cf. [53, Théorème 5.4]). Assume that G is of compact type
e
and that δe < ∞. Then there exists a δ-quasiconformal
measure supported on Λ. If
e
X is strongly hyperbolic, then there exists a δ-conformal
measure supported on Λ.

Remark 15.4.7. Any group acting on a proper geodesic hyperbolic metric

space is of compact type, so Theorem 15.4.6 includes the case of proper geodesic
hyperbolic metric spaces.
Remark 15.4.8. Combining Theorem 15.4.6 with Proposition 15.4.4 and Corollary 15.4.3 shows that for G nonelementary of compact type,
δe = inf{s > 0 : there exists an s-quasiconformal measure supported on Λ},

thus giving another geometric characterization of δe (the first being Theorem 1.2.3).

Before proving Theorem 15.4.6, we recall the following lemma due to Patterson:

15.4. QUASICONFORMAL MEASURES

263

Lemma 15.4.9 ([142, Lemma 3.1]). Let A = (an )∞
1 be a sequence of positive

real numbers, and let

δ = δ(A) = inf

(

s≥0:

∞
X

a−s
n

n=1

)

<∞ .

Then there exists an increasing continuous function k : (0, ∞) → (0, ∞) such that:
(i) The series
Σs,k (A) =

∞
X

k(an )a−s
n

n=1

converges for s > δ and diverges for s ≤ δ.

(ii) There exists a decreasing function ε : (0, ∞) → (0, ∞) such that for all
y > 0 and x > 1,
k(xy) ≤ xε(y) k(y),

(15.4.1)

and such that limy→∞ ε(y) = 0.
Proof of Theorem 15.4.6. By Proposition 8.2.4, there exist ρ > 0 and a
e
maximal ρ-separated set Sρ ⊆ G(o) such that δ(G)
= δ(Sρ ); moreover, this ρ

may be chosen large enough so that Sρ/2 does not contain a bounded infinite set,
where Sρ/2 is a ρ/2-separated set. Let A = (an )∞
1 be any indexing of the sequence

(bkxk )x∈Sρ , and let k : (0, ∞) → (0, ∞) be the function given by Lemma 15.4.9. For
shorthand let
k(x) = k(bkxk )
ε(x) = ε(bkxk )
Σs,k = Σs,k (A) =

X

k(x)b−skxk .

x∈Sρ

e moreover, the function s 7→ Σs,k is continuous.
Then Σs,k < ∞ if and only if s > δ;
For each s > δeG , let
1 X
(15.4.2)
µs =
k(x)b−skxk δx ∈ M1 (Sρ ∪ Λ).
Σs,k
x∈Sρ

Now since G is of compact type, the set Sρ ∪ Λ is compact (cf. (B) of Proposition

7.7.2). Thus by the Banach–Alaoglu theorem, the set M1 (Sρ ∪ Λ) is compact in
the weak-* topology. So there exists a sequence sn ց δe so that if we let µn = µsn ,
then µn → µ ∈ M1 (Sρ ∪ Λ). We will show that µ is δeG -quasiconformal and that

Supp(µ) = Λ.

Claim 15.4.10. Supp(µ) ⊆ Λ.

264

15. CONFORMAL AND QUASICONFORMAL MEASURES

Proof. Fix R > 0. Since δ(Sρ ) < ∞, we have #(Sρ ∩ B(o, R)) < ∞. Thus,
µ(B(o, R)) ≤ lim sup µs (B(o, R)) ≤ lim sup
e
sցδ

e
sցδ

=

e

#(Sρ ∩ B(o, R))k(bR )b−δR
Σs,k
e

#(Sρ ∩ B(o, R))k(bR )b−δR
= 0.
∞

Letting R → ∞ shows that µ(X) = 0; thus Supp(µ) ⊆ Sρ ∪ Λ \ X = Λ.

⊳

e
To complete the proof, we must show that µ is δ-quasiconformal.
Fix g ∈ G,

and let

e

νg = [(g ′ )δ µ] ◦ g −1 .

We want to show that νg ≍× µ.
Claim 15.4.11. For every continuous function f : bord X → (0, ∞), we have
Z
Z
(15.4.3)
f dνg ≍× f dµ.
Proof. Since Sρ ∪ Λ is compact, logb (f ) is uniformly continuous on Sρ ∪ Λ
with respect to the metric D. Let φf denote the modulus of continuity of logb (f ),
so that
f (x)
≤ bφf (r) ∀x, y ∈ Sρ ∪ Λ.
f (y)

D(x, y) ≤ r ⇒

(15.4.4)

For each n ∈ N let

νg,n = [(g ′ )sn µn ] ◦ g −1 ,

so that νg,n −−→ ν. Then
n,×

νg,n =
≍×
=
=
and so
(15.4.5)

1
Σsn ,k
1
Σsn ,k
1
Σsn ,k
1
Σsn ,k

X

x∈Sρ

X

x∈Sρ

X

k(x)b−sn kxk [(g ′ )sn δx ] ◦ g −1
k(x)b−sn kxk bsn [kxk−kg(x)k] [δx ◦ g −1 ]
b−sn kg(x)k k(x)δg(x)

x∈Sρ

X

b−sn kxk k(g −1 (x))δx ,

x∈g(Sρ )

P
R
−sn kxk
k(g −1 (x))f (x)
f dνg,n
x∈g(Sρ ) b
R
P
·
≍×
−sn kyk k(y)f (y)
f dµn
y∈Sρ b

For each x ∈ g(Sρ ) ⊆ G(o), there exists yx ∈ Sρ such that d(x, yx ) ≤ ρ.
Observation 15.4.12. #{x : yx = y} is bounded independent of y and g.

15.4. QUASICONFORMAL MEASURES

265

Proof. Write y = h(o); then
#{x : yx = y} ≤ #(g(Sρ ) ∩ B(y, ρ)) = #(h−1 g(Sρ ) ∩ B(o, ρ)).
But Sρ′ := h−1 g(Sρ ) is a ρ-separated set. For each x ∈ Sρ′ , choose zx ∈ Sρ/2 such
that d(x, zx ) < ρ/2; then the map x 7→ zx is injective, so
#(Sρ′ ) ≤ #(Sρ/2 ∩ B(o, 2ρ)),
which is bounded independent of y and g.

⊳

Now
D(x, yx ) ≤ b−hx|yxio ≤ bρ−kyx k ;
applying (15.4.4) gives
ρ−kyx k
)

f (x) ≤ bφf (b

f (yx ).

On the other hand, by (15.4.1) we have
k(g −1 (x)) ≤ bε(yx )[ρ+kgk] k(yx ),
and we also have
b−sn kxk ≤ bsn ρ b−sn kyx k .
Combining everything gives
X
b−sn kxk k(g −1 (x))f (x)
x∈g(Sρ )

≤
.×

X

x∈g(Sρ )

X

y∈Sρ



expb sn ρ + ε(yx )[ρ + kgk] + φf (bρ−kyx k ) b−sn kyx k k(yx )f (yx )



expb ε(y)[ρ + kgk] + φf (bρ−kyk ) b−sn kyk k(y)f (y),

and taking the limit as n → ∞ we have
Z
Z
Z


f (x) dν(x) .× expb ε(y)[ρ + kgk] + φf (bρ−kyk ) f (y) dµ(y) = f (y) dµ(y)

since φf (bρ−kyk ) = ε(y) = 0 for all y ∈ ∂X. A symmetric argument gives the
converse direction.

⊳

Now let C be the implied constant of (15.4.3). Then for every continuous
function f : X → (0, ∞),
Z
Z
Z
Z
C f dν − f dµ ≥ 0 and C f dµ − f dν ≥ 0,

R
R
R
R
i.e. the linear functionals I1 [f ] = C f dν − f dµ and I2 [f ] = C f dµ − f dν
are positive. Thus by the Riesz representation theorem, there exist measures γ1 , γ2

such that Iγi = Ii (i = 1, 2). The uniqueness assertion of the Riesz representation

266

15. CONFORMAL AND QUASICONFORMAL MEASURES

theorem then guarantees that
(15.4.6)

γ1 + µ = Cν and γ2 + ν = Cµ.

In particular, Cν ≥ µ, and Cµ ≥ ν. This completes the proof.



CHAPTER 16

Patterson–Sullivan theorem for groups of
divergence type
In this chapter, we prove Theorem 1.4.1, which states that a nonelementary
e
measure.
group of generalized divergence type possesses a δ-quasiconformal
16.1. Samuel–Smirnov compactifications

We begin by summarizing the theory of Samuel–Smirnov compactifications,
which will be used in the proof of Theorem 1.4.1.
Proposition 16.1.1. Let (Z, D) be a complete metric space. Then there exists
b together with a homeomorphic embedding ι : Z → Z
b
a compact Hausdorff space Z
with the following property:
Property 16.1.2. If A, B ⊆ Z, then A ∩ B 6=  if and only if
D(A, B) = 0. Here A and B denote the closures of A and B relative
b
to Z.

b ι) is unique up to homeomorphism. Moreover, if Z1 , Z2 are two comThe pair (Z,
plete metric spaces and if f : Z1 → Z2 is uniformly continuous, then there exists
b1 → Z
b2 such that ι ◦ f = fb ◦ ι. The reverse is also
a unique continuous map fb : Z
true: if f admits such an extension, then f is uniformly continuous.

b will be called the Samuel–Smirnov compactification of Z.
The space Z

Proof of Proposition 16.1.1. The metric D induces a proximity on Z in
the sense of [136, Definition 1.7]. Then the existence and uniqueness of a pair
b ι) for which Property 16.1.2 holds is guaranteed by [136, Theorem 7.7]. The
(Z,

assertions concerning uniformly continuous maps follow from [136, Theorem 7.10]
and [136, Theorem 4.4], respectively (cf. [136, Remark 4.8] and [136, Definition
4.10]).



Remark 16.1.3. The Samuel–Smirnov compactification may be compared with
the Stone–Čech compactification, which is usually larger. The difference is that
instead of Property 16.1.2, the Stone–Čech compactification has the property that
for all A, B ⊆ Z, A ∩ B 6=  if and only if A ∩ B ∩ Z 6= . Moreover, in the
267

268

16. PATTERSON–SULLIVAN THEOREM FOR GROUPS OF DIVERGENCE TYPE

remarks following Property 16.1.2, “uniformly continuous” should be replaced with
just “continuous”.
We remark that if δG < ∞ (i.e. if G is of divergence type rather than of

generalized divergence type), then the proof below works equally well if the Samuel–
Smirnov compactification is replaced by the Stone–Čech compactification. This is
not the case for the general proof; cf. Remark 16.3.5.

To prove Theorem 1.4.1, we will consider the Samuel–Smirnov compactification
of the complete metric space (bord X, D) (cf. Proposition 3.6.13), which we will
b For convenience of notation we will assume that bord X is a subset
denote by X.
b and that ι : bord X → X
b is the inclusion map. As a point of terminology we
of X

b \ bord X “nonstandard
will call points in bord X “standard points” and points in X
points”.

Remark 16.1.4. Since D ≍× Dx for all x ∈ X, the Samuel–Smirnov compactb is independent of the basepoint o.
ification X

At this point we can give a basic outline of the proof of Theorem 1.4.1: First
b which satisfies the transformation equation
we will construct a measure µ
b on X

(15.1.1). We will call such a measure µ
b a quasiconformal measure, although it is
not a priori a quasiconformal measure in the sense of Definition 15.1.1, as it is not

necessarily supported on the set of standard points. Then we will use Thurston’s
proof of the Hopf–Tsuji–Sullivan theorem [4, Theorem 4 of Section VII] (see also
[138, Theorem 2.4.6]) to show that µ
b is supported on the nonstandard analogue of

radial limit set. Finally, we will show that the nonstandard analogue of the radial
limit set is actually a subset of bord X, i.e. we will show that radial limit points
are automatically standard. This demonstrates that µ
b is a measure on bord X, and
is therefore a bona fide quasiconformal measure.

We now begin the preliminaries to the proof of Theorem 1.4.1. As always
b be the Samuel–Smirnov compactification
(X, o, b) denotes a Gromov triple. Let X
of bord X.

Remark 16.1.5. Throughout this chapter, S denotes the closure of a set S
b not bord X.
taken with respect to X,
b
16.2. Extending the geometric functions to X

We begin by extending the geometric functions d(·, ·), h·|·i, and B(·, ·) to the
b Extending d(·, ·) is the easiest:
Samuel–Smirnov compactification X.

Observation 16.2.1. If x ∈ X is fixed, then the function fx : bord X → [0, 1]

defined by fx (y) = b−d(x,y) is uniformly continuous by Remark 3.6.15. Thus by

b
16.2. EXTENDING THE GEOMETRIC FUNCTIONS TO X

269

b → [0, 1]. We
Proposition 16.1.1, there exists a unique continuous extension fbx : X

write

b yb) = − logb fbx (b
d(x,
y ).

We define the extended boundary of X to be the set

d := {b
b : d(o,
b b
∂X
ξ∈X
ξ) = ∞}.

d ∩ bord X = ∂X.
b y) = d(x, y) if x, y ∈ X, and ∂X
Note that d(x,
d 6= ∂X.
Warning. It is possible that ∂X

b presents some diffiOn the other hand, extending the Gromov product to X
culty, since the Gromov product is not necessarily continuous (cf. Example 3.4.6).

Our solution is as follows: Fix x ∈ X and y ∈ bord X. Then by Remark 3.6.15,
the map bord X ∋ z 7→ Dx (y, z) is uniformly continuous, so by Proposition 16.1.1
b ∋ zb 7→ D
b x (y, zb). We define the Gromov product
it extends to a continuous map X
b via the formula
in X
b x (y, zb).
hy|b
z ix = − logb D

Note that if zb ∈ bord X, then this notation conflicts with the previous definition
of the Gromov product, but by Proposition 3.6.8 the harm is only an additive
asymptotic. We will ignore this issue in what follows.

Observation 16.2.2. Using (j) of Proposition 3.3.3 we may define for each
x, y ∈ X the Busemann function
b zb(x, y) = hx|b
B
z iy − hy|b
z ix .

Again, if zb ∈ bord X, then this definition conflicts with the previous one, but again

the harm is only an additive asymptotic.

Remark 16.2.3. We note that an appropriate analogue of Proposition 3.3.3
b Specifically, each formula of Proposition
(cf. also Corollary 3.4.12) holds on X.
3.3.3 holds with an additive asymptotic, as long as all expressions are defined. Note
in particular that we have not defined the value of expressions which contain more
than one nonstandard point. Such a definition would present additional difficulties
(namely, noncommutativity of limits) which we choose to avoid.
We are now ready to define the nonstandard analogue of the radial limit set:

let

Definition 16.2.4 (cf. Definitions 4.5.1 and 7.1.2). Given x ∈ X and σ > 0,
b : ho|b
[
Shad(x,
σ) = {b
ξ∈X
ξix ≤ σ},

[
so that Shad(x,
σ) ∩ bord X = Shad(x, σ). A sequence (xn )∞
1 in X will be said to
d
b
[ n , σ) for all
converge to a point ξ ∈ ∂X σ-radially if kxn k → ∞ and if b
ξ ∈ Shad(x

270

16. PATTERSON–SULLIVAN THEOREM FOR GROUPS OF DIVERGENCE TYPE

n ∈ N. Note that in the definition of σ-radial convergence, we do not require that
b although this can be seen from the proof of Lemma
xn → b
ξ in the topology on X,
16.2.5 below.
We conclude this section with the following lemma:
Lemma 16.2.5 (Every radial limit point is a standard point). Suppose that a
d
b
sequence (xn )∞
1 converges to a point ξ ∈ ∂X σ-radially for some σ > 0. Then
b
ξ ∈ ∂X.
Proof. We observe first that

b b
→ d(o,
ξ) = ∞.
hxn |b
ξio ≍+ kxn k − ho|b
ξixn ≍+,σ kxn k −
n

Together with Gromov’s inequality hxn |xm io &+ min(hxn |b
ξio , hxm |b
ξio ), this implies

that (xn )∞
1 is a Gromov sequence.
By the definition of the Gromov boundary, it follows that there exists a (standard) point η ∈ ∂X such that the sequence (xn )∞
1 converges to η. Gromov’s
b
inequality now implies that hη|ξio = ∞. We claim now that b
ξ = η, so that b
ξ is
b is a Hausdorff space, it follows
standard. By contradiction, suppose b
ξ 6= η. Since X
b containing b
that there exist disjoint open sets U, V ⊆ X
ξ and η, respectively. Since

V contains a neighborhood of η, the function fo,η (z) = hη|zio is bounded from
above on bord X \ V . By continuity, fbo,η is bounded from above on bord X \ V . In
particular, b
ξ∈
/ bord X \ V . On the other hand b
ξ∈
/ V , since b
ξ is in the open set U
b a contradiction.
which is disjoint from V . It follows that b
ξ∈
/ bord X = X,



Remark 16.2.6. In fact, the above proof shows that if

(16.2.1)

hxn |b
ξio → ∞

b d
b
for some sequence (xn )∞
1 in X and some ξ ∈ ∂X, then ξ ∈ ∂X. However, there
∞
b but for which
may be a sequence (xn )1 such that xn → b
ξ in the topology on X
b
(16.2.1) does not hold. In this case, we could have ξ ∈
/ ∂X.
b
16.3. Quasiconformal measures on X

b as follows:
We define the notion of a quasiconformal measure on X

Definition 16.3.1 (cf. Definition 15.1.1, Proposition 4.2.6). For each s ≥ 0, a
d is called s-quasiconformal if
Radon probability measure µ
b on ∂X
Z
−1
b
µ
b(b
g (A)) ≍×
bsBηb (o,g (o)) db
µ(b
η ).
A

d Here b
for every g ∈ G and for every Borel set A ⊆ ∂X.
g denotes the unique
b (cf. Proposition 16.1.1).
continuous extension of g to X

b
16.3. QUASICONFORMAL MEASURES ON X

271

Remark 16.3.2. Note that we have added here the assumption that the measure µ
b is Radon. Since the phrase “Radon measure” seems to have no generally
accepted meaning in the literature, we should make clear that for us a (finite,
nonnegative, Borel) measure µ on a compact Hausdorff space Z is Radon if the
following two conditions hold (cf. [74, §7]):
µ(A) = inf{µ(U ) : U ⊇ A, U open} ∀A ⊆ Z Borel
µ(U ) = sup{µ(K) : K ⊆ U, K compact} ∀U ⊆ Z open.
The assumption of Radonness was not needed in Definition 15.1.1, since every
measure on a compact metric space is Radon [74, Theorem 7.8]. However, the
b is not necessarily metrizable,
assumption is important in the present proof, since X

and so it may have non-Radon measures.
On the other hand, the Radon condition itself is of no importance to us, except
for the following facts:
(i) The image of a Radon measure under a homeomorphism is Radon.
(ii) Every measure absolutely continuous to a Radon measure is Radon.

(iii) The sum of two Radon measures is Radon.
(iv) (Riesz representation theorem, [74, Theorem 7.2]) Let Z be a compact
Hausdorff space. For each measure µ on Z, let Iµ denote the nonnegative
linear function
Z
Iµ [f ] := f dµ.
Then for every nonnegative linear functional I : C(Z) → R, there exists

a unique Radon measure µ on Z such that Iµ = I. (If µ1 and µ2 are not
both Radon, it is possible that Iµ1 = Iµ2 while µ1 6= µ2 .)

We now state two lemmas which are nonstandard analogues of lemmas proven
in Chapter 15. We omit the parts of the proofs which are the same as in the
standard case, reminding the reader that the important point is that no function is
ever used which takes two nonstandard points as inputs. We begin by proving an
analogue of Sullivan’s shadow lemma:

b cf. Lemma 15.4.1). Fix s ≥ 0,
Lemma 16.3.3 (Sullivan’s Shadow Lemma on X;
d
and let µ
b ∈ M(∂X) be an s-quasiconformal measure which is not a pointmass

supported on a standard point. Then for all σ > 0 sufficiently large and for all
g ∈ G, we have
[
µ
b(Shad(g(o),
σ)) ≍× b−skgk .

272

16. PATTERSON–SULLIVAN THEOREM FOR GROUPS OF DIVERGENCE TYPE

Proof. Obvious modifications1 to the proof of Lemma 15.4.1 yield

[
[ g−1 (o) (o, σ) .
µ
b(Shad(g(o),
σ)) ≍×,µ,σ b−skgk µ
b Shad

So to complete the proof, we need to show that

[ g−1 (o) (o, σ) ≍×,µ,σ 1,
µ
b Shad

assuming σ > 0 is sufficiently large (depending on µ
b). By contradiction, suppose
that for each n ∈ N there exists gn ∈ G such that

[ −1 (o, n) ≤ 1 ·
µ
b Shad
gn (o)
2n
b
Then for µ
b-a.e. b
ξ ∈ X,
which implies

b
[ −1 (o, n) for all but finitely many n,
ξ ∈ Shad
gn (o)
hgn−1 (o)|b
ξio &+ n −
→ ∞.
n

By Remark 16.2.6, it follows that b
ξ ∈ ∂X and gn−1 (o) → b
ξ. This implies that µ
b

is a pointmass supported on the standard point limn→∞ gn−1 (o), contradicting our
hypothesis.

Lemma 16.3.4 (cf. Theorem 15.4.6). Assume that δe = δeG < ∞. Then there
e
d
exists a δ-quasiconformal
measure supported on ∂X.

b replaces
Proof. Let the measures µs be as in (15.4.2). The compactness of X
the assumption that G is of compact type which occurs in Theorem 15.4.6, so
b for some Radon measure
there exists a sequence sn ց δe such that µn := µsn → µ
d
b
µ
b ∈ M(X). Claim 15.4.10 shows that µ
b is supported on ∂X.
e
To complete the proof, we must show that µ
b is δ-quasiconformal.
Fix g ∈ G and
b → (0, ∞). The final assertion of Proposition 16.1.1
a continuous function f : X

guarantees that log(f ) ↿ bord X is uniformly continuous, so the proof of Claim
15.4.11 shows that (15.4.3) holds.
The equation (15.4.6) deserves some comment; it depends on the uniqueness

assertion of the Riesz representation theorem, which, now that we are no longer in
a metric space, holds only for Radon measures. But by Remark 16.3.2, all measures
involved in (15.4.6) are Radon, so (15.4.6) still holds.



Remark 16.3.5. In this lemma we used the final assertion of Proposition 16.1.1
in a nontrivial way. The proof of this lemma would not work for the Stone–Čech
1We remark that the expression g ′ (ξ) occuring in the proof of Lemma 15.4.1 should be replaced
b (o,g −1 (o))
−B

b makes no sense, since
by b ξb
as per Proposition 4.2.6; of course, the expression g ′ (ξ)
b is not a metric space.
X

16.4. THE MAIN ARGUMENT

273

compactification, except in the case δ < ∞, in which case the uniform continuity
of f is not necessary in the proof of Theorem 15.4.6.

b cf. Lemma 4.5.4). For
Lemma 16.3.6 (Intersecting Shadows Lemma on X;
each σ > 0, there exists τ = τσ > 0 such that for all x, y, z ∈ X satisfying d(z, y) ≥
[ z (x, σ) ∩ Shad
[ z (y, σ) 6= , we have
d(z, x) and Shad
(16.3.1)

[ z (y, σ) ⊆ Shad
[ z (x, τ )
Shad

and
(16.3.2)

d(x, y) ≍+,σ d(z, y) − d(z, x).

Proof. The proof of Lemma 4.5.4 goes through with no modifications needed.

16.4. The main argument
Proposition 16.4.1 (Generalization/nonstandard version of Theorem 1.4.2(A)
e
d which is not a pointmass sup⇒ (B)). Let µ
b be a δ-quasiconformal
measure on ∂X
ported on a standard point. If G is of generalized divergence type, then µ
b(Λr (G)) >

0.

Proof. Fix σ > 0 large enough so that Sullivan’s Shadow Lemma 16.3.3 holds.
Let ρ > 0 be large enough so that there exists a maximal ρ-separated set Sρ ⊆ G(o)
which has finite intersection with bounded sets (cf. Proposition 8.2.4(iii)). Let
(xn )∞
1 be an indexing of Sρ . By Lemma 16.2.5, we have
\ [
[ n , σ + ρ) ⊆ Λr (G).
Shad(x
N ∈N n≥N

By contradiction suppose that µ
b(Λr (G)) = 0. Fix ε > 0 small to be determined.

Then there exists N ∈ N such that


[
[ n , σ + ρ) ≤ ε.
µ
b
Shad(x
n≥N

Let R = ρ + maxn<N kxn k. Then




 [

[
µ
b
Shad(g(o),
σ)

 ≤ ε.
g∈G
kgk>R

We shall prove the following.

Observation 16.4.2. If A ⊆ G(o) is any subcollection satisfying
(I) kxk > R for all x ∈ A, and

274

16. PATTERSON–SULLIVAN THEOREM FOR GROUPS OF DIVERGENCE TYPE

[
(II) (Shad(x,
σ))x∈A are disjoint,
then
X

(16.4.1)

b−skxk .× ε.

x∈A

Proof. The disjointness condition guarantees that

X

x∈A



 [

[
[ n (o), σ) ≤ ε.
µ
b(Shad(x,
σ)) ≤ µ
b
Shad(g


g∈G
kgk>R

Combining with Sullivan’s Shadow Lemma 16.3.3 yields (16.4.1).

⊳

Now choose R′ > R and σ ′ > σ large to be determined. Let SR′ be a maximal
′

R -separated subset of G(o). For convenience we assume o ∈ SR′ . By Proposition
8.2.4(iv), if R′ is sufficiently large then Σδe(SR′ ) = ∞ if and only if δe is of generalized

divergence type. So to complete the proof, it suffices to show that
Σδe(SR′ ) < ∞.

Notation 16.4.3. Let (xi )∞
1 be an indexing of SR′ such that i < j implies
kxi k ≤ kxj k. For xi , xj ∈ SR′ distinct, we write xi < xj if
(I) i < j and
[ i , σ ′ ) ∩ Shad(x
[ j , σ ′ ) 6= .
(II) Shad(x
(This is just a notation, it does not mean that < is a partial order on SR′ .)
Lemma 16.4.4. If R′ and σ ′ are sufficiently large (with σ ′ chosen first), then
[ x (y, σ) ⊆ Shad(y,
[
x < y ⇒ Shad
σ ′ ).
[
[ σ ′ ) 6= . By the Intersecting
Proof. Suppose x < y; then Shad(x,
σ ′ )∩Shad(y,
Shadows Lemma 16.3.6, we have d(x, y) ≍+,σ′ kyk − kxk. On the other hand, since
SR′ is R′ -separated we have d(x, y) ≥ R′ . Thus

ho|xiy &+,σ′ R′ .
b we have
Now for any b
ξ ∈ X,

hx|b
ξiy &+ min(ho|b
ξiy , ho|xiy ).

[ x (y, σ), then
Thus if b
ξ ∈ Shad

σ &+ ho|b
ξiy or σ &+,σ′ R′ .

Let σ ′ be σ plus the implied constant of the first asymptotic, and then let R′ be σ+1
plus the implied constant of the second asymptotic. Then the second asymptotic

16.4. THE MAIN ARGUMENT

275

is automatically impossible, so
ho|b
ξiy ≤ σ ′ ,

[
i.e. b
ξ ∈ Shad(y,
σ ′ ).

⊳

If x ∈ SR′ is fixed, let us call y ∈ SR′ an immediate successor of x if x < y

but there is no z such that x < z < y. We denote by SR′ (x) the collection of all
immediate successors of x.
Lemma 16.4.5. For each z ∈ SR′ , we have
X
b−skyk .× εb−skzk .
(16.4.2)
y∈SR′ (z)

[ σ ′ ))y∈S ′ (z) consists of muProof. We claim first that the collection (Shad(y,
R
[ 1 , σ ′ ) ∩ Shad(y
[ 2 , σ ′ ) 6=  for some distinct
tually disjoint sets. Indeed, if Shad(y

y1 , y2 ∈ SR′ (z), then we would have either z < y1 < y2 or z < y2 < y1 , contradicting the definition of immediate successor. Combining with Lemma 16.4.4, we see
[ z (y, σ))y∈S ′ (z) also consists of mutually disjoint sets.
that the collection (Shad
R

Fix g ∈ G such that g(o) = z. We claim that the collection
A = g −1 (SR′ (z))
satisfies the hypotheses of Observation 16.4.2. Indeed, as o ∈
/ A (since z ∈
/ SR′ (z))

and as g is an isometry of X, (I) follows from the fact that SR′ is R′ -separated and
[ −1 (y), σ) = g −1 (Shad
[ z (y, σ)) for all y ∈ SR′ (z), the collection
R′ > R. Since Shad(g
[
(Shad(x,
σ))x∈A consists of mutually disjoint sets, meaning that (II) holds. Thus,
by Observation 16.4.2, we have
X

b−skxk .× ε,

x∈A

or, since g is an isometry of X and z = g(o),
X
b−sd(z,y) .× ε.
y∈SR′ (z)

Inserting (16.3.2) into the last inequality yields (16.4.2).

⊳

Using Lemma 16.4.5, we complete the proof. Define the sequence (Sn )∞
n=0
inductively as follows:
S0 = {o},
[
Sn+1 =
SR′ (x).
x∈Sn

276

16. PATTERSON–SULLIVAN THEOREM FOR GROUPS OF DIVERGENCE TYPE

Clearly, all immediate successors of all points of
claim that
S R′ =

[

S

n≥0

Sn belong to

S

n≥0

Sn . We

Sn .

n≥0

Indeed, let (xi )∞
1 be the indexing of SR′ considered in Notation 16.4.3, and by
S∞
induction suppose that xi ∈ 1 Sn for all i < j. If j = 0, then xj = o ∈ S0 .
Otherwise, let i < j be maximal satisfying xi < xj . Then xj is an immediate
S∞
S∞
successor of xi ∈ 1 Sn , so xj ∈ 1 Sn .
Summing (16.4.2) over all x ∈ Sn , we have
X
X
b−skyk .× ε
b−skxk .
y∈Sn+1

x∈Sn

Set ε equal to 1/2 divided by the implied constant, so that
X
1 X −skxk
b−skyk ≤
b
.
2
y∈Sn+1

x∈Sn

Applying the Ratio Test, we see that the series Σδe(SR′ ) converges, contradicting

that G was of generalized divergence type.



e
d If G is of
Corollary 16.4.6. Let µ
b be a δ-quasiconformal
measure on ∂X.

generalized divergence type, then µ
b(Λr (G)) = 1.

d \ Λr (G) is a δe
Proof. By contradiction suppose not. Then νb := µ
b ↿ ∂X
d which gives zero measure to Λr (G), contradicting
quasiconformal measure on ∂X

Proposition 16.4.1.



16.5. End of the argument
We now complete the proof of Theorem 1.4.1:
e
Proof of Theorem 1.4.1. Let µ
b be the δ-quasiconformal
measure supported
d
on ∂X guaranteed by Lemma 16.3.4. By Corollary 15.4.3, µ
b is not a pointmass

supported on a standard point. By Corollary 16.4.6, µ
b is supported on Λr (G) ⊆ ∂X.
This completes the proof of the existence assertion.
e
Suppose that µ1 , µ2 are two δ-quasiconformal
measures on ∂X. By Corollary

16.4.6, µ1 and µ2 are both supported on Λr (G).

Suppose first that µ1 , µ2 are supported on Λr,σ for some σ > 0. Fix an open
set U ⊆ ∂X. By the Vitali covering theorem, there exists a collection of disjoint
shadows (Shad(g(o), σ))g∈A contained in U such that
[
µ1 (U \
Shad(g(o), σ)) = 0.
g∈A

16.6. NECESSITY OF THE GENERALIZED DIVERGENCE TYPE ASSUMPTION

277

Then we have
µ1 (U ) =

X

g∈A

µ1 (Shad(g(o), σ)) ≍×,µ1
≍×,µ2

X

b−skgk

g∈A

X

µ2 (Shad(g(o), σ))

g∈A

≤ µ2 (U ).
A similar argument shows that µ2 (U ) .× µ1 (U ). Since U was arbitrary, a standard
approximation argument shows that µ1 ≍× µ2 . It follows that any individual
measure µ supported on Λr,σ is ergodic, because if A is an invariant set with 0 <
1
1
µ ↿ A and 1−µ(A)
µ ↿ (Λr \ A) are two measures which are not
µ(A) < 1 then µ(A)
asymptotic, a contradiction.
In the general case, define the function f : Λr → [0, ∞) by
f (ξ) = sup{σ > 0 : ∃g ∈ G g(ξ) ∈ Λr,σ }.
By Proposition 7.2.3, f (ξ) < ∞ for all ξ ∈ Λr . On the other hand, f is G-invariant.
e
Now let µ be a δ-quasiconformal
measure on Λr . Then for each σ0 < ∞ the measure
µ ↿ f −1 ([0, σ0 ]) is supported on Λr,σ0 , and is therefore ergodic; thus f is constant

µ ↿ f −1 ([0, σ0 ])-a.s. It is clear that this constant value is independent of σ0 for large
enough σ0 , so f is constant µ-a.s. Thus there exists σ > 0 such that µ is supported
on Λr,σ , and we can reduce to the previous case.



16.6. Necessity of the generalized divergence type assumption
The proof of Theorem 1.4.1 makes crucial use of the generalized divergence type
assumption, just as the proof of Theorem 15.4.6 made crucial use of the compact
type assumption. What happens if neither of these assumptions holds? Then there
e
may not be a δ-quasiconformal
measures supported on the limit set, as we now

show:

Proposition 16.6.1. There exists a strongly discrete group of general type
G ≤ Isom(H∞ ) satisfying δ < ∞, such that there does not exist any quasiconformal

measure supported on Λ.

Proof. The idea is to first construct such a group in an R-tree, and then to
use a BIM embedding (Theorem 13.1.1) to get an example in H∞ . Fix a sequence
of numbers (ak )∞
1 . For each k let Γk = {e, γk } ≡ Z2 , and let k · k : Γk → R be
defined by kγk k = ak , kek = 0. Clearly, the function k · k is tree-geometric in the

sense of Definition 14.5.2, so by Theorem 14.5.5, the function k · k : Γ → [0, ∞)

defined by (14.5.1) is tree-geometric, where Γ = ∗k∈N Γk . So there exist an R-tree
X and a homomorphism φ : Γ → Isom(X) such that kφ(γ)k = kγk ∀γ ∈ Γ. Let

G = φ(Γ).

278

16. PATTERSON–SULLIVAN THEOREM FOR GROUPS OF DIVERGENCE TYPE

Claim 16.6.2. If the sequences (ak )∞
1 is chosen appropriately, then G is of
convergence type.
Proof. For s ≥ 0 we have
X
e−skgk
Σs (G) − 1 =
g∈G\{id}

=

X

(k1 ,γ1 )···(kn ,γn )∈(ΓE )∗ \{∅}

=

X

X

X

X

X



exp − s ak1 + . . . + akn

n∈N k1 6=k2 6=···6=kn γ1 ∈Γk1 \{e}

=

n
Y

n∈N k1 6=k2 6=···6=kn i=1

Σs (G) ≤ 1 +
Σs (G) ≥ 1 +

X

n∈N

X

X

e

−sak

k∈N

···

X

γn ∈Γkn \{e}



exp − s ak1 + . . . + akn

e−saki
!n

e−sak .

k∈N

Thus, letting
Ps =

X

e−sak ,

k∈N

we have
(16.6.1)


Σ (G) < ∞
s
Σ (G) = ∞
s

if Ps < 1
if Ps = ∞

.

Now clearly, there exists a sequence (ak )∞
1 such that P1/2 < 1 but Ps = ∞ for all

s < 1/2; for example, take ak = log(k) + 2 log log(k) + C for sufficiently large C.
⊳
Claim 16.6.3. Λ(G) = Λr (G).

Proof. For all ξ ∈ Λ, the path traced by the geodesic ray [o, ξ] in X/G is the
S
concatenation of infinitely many paths of the form [o, g(o)], where g ∈ n∈N φ(Γn ).
Each such path crosses o, so the path traced by the geodesic ray [o, ξ] in X/G
crosses o infinitely often. Equivalently, the geodesic ray [o, ξ] crosses G(o) infinitely
often. By Proposition 7.1.1, this implies that ξ ∈ Λr (G).
⊳
e be the image of G under a BIM representation (cf. Theorem 13.1.1).
Now let G
e is of convergence type and Λ(G)
e = Λr (G).
e
By Remark 13.1.4, G
The proof is
completed by the following lemma:

Lemma 16.6.4. If the group G is of generalized convergence type and µ is a
e
δ-quasiconformal
measure, then µ(Λr ) = 0.

16.7. ORBITAL COUNTING FUNCTIONS OF NONELEMENTARY GROUPS

279

Proof. Fix σ > 0 large enough so that Sullivan’s Shadow Lemma 15.4.1 holds.
Fix ρ > 0 and a maximal ρ-separated set Sρ ⊆ G(o) such that Σδe(Sρ ) < ∞. Then
X
X e
µ(Shad(x, ρ + σ)) ≍×,ρ,σ
b−δkxk < ∞.
x∈Sρ

x∈Sρ

On the other hand, Λr,σ ⊆ lim supx∈Sρ Shad(x, ρ + σ). So by the Borel–Cantelli
lemma, µ(Λr,σ ) = 0. Since σ was arbitrary, µ(Λr ) = 0.
⊳

Combining Theorem 1.4.1 and Lemma 16.6.4 yields the following:
Proposition 16.6.5. Let G ≤ Isom(X) be a nonelementary group with δe < ∞.

Then the following are equivalent:

(A) G is of generalized divergence type.
e
(B) There exists a δ-conformal
measure µ on Λ satisfying µ(Λr ) > 0.
e
(C) Every δ-conformal
measure µ on Λ satisfies µ(Λr ) = 1.
e
(D) There exists a unique δ-conformal
measure µ on Λ, and it satisfies µ(Λr ) =
1.

16.7. Orbital counting functions of nonelementary groups
Theorem 1.4.2 allows us to prove the following result which, on the face of it,
does not involve quasiconformal measures at all:
Corollary 16.7.1. Let G ≤ Isom(X) be nonelementary and satisfy δ < ∞.
Then
NX,G (ρ) .× bδρ ∀ρ ≥ 0.
Proof. If G is of convergence type, then the bound is obvious, as
X
b−δρ NX,G (ρ) ≤
b−δkgk ≤ Σδ (G) < ∞.
g∈G
kgk≤ρ

On the other hand, if G is of divergence type, then by Theorem 1.4.1, there exists
a δ-conformal measure µ on Λ, which is not a pointmass by Corollary 15.4.3 and
Proposition 10.5.4(C). Remark 15.4.5 finishes the proof.

We contrast this with a philosophically related result whose proof uses the
Ahlfors regular measures constructed in the proof of Theorem 1.2.1:
Proposition 16.7.2. Let G ≤ Isom(X) be a nonelementary and strongly dis-

crete. Then

logb NX,G (ρ)
−−−→ δ.
ρ→∞
ρ

280

16. PATTERSON–SULLIVAN THEOREM FOR GROUPS OF DIVERGENCE TYPE

Proof. Fix s < δ. By Theorem 1.2.1, there exist τ > 0 and an Ahlfors sregular measure µ supported on Λur,τ . Now fix ρ > 0. By definition, for all ξ ∈ Λur,τ
there exists g ∈ G such that ρ − τ ≤ kgk ≤ ρ and ho|ξig(o) ≤ τ . Equivalently,
[
Λur,τ ⊆
Shad(g(o), τ ).
g∈G
ρ−τ ≤kgk≤ρ

Applying µ to both sides gives
1≤

X

g∈G
ρ−τ ≤kgk≤ρ


µ Shad(g(o), τ ) .

By the Diameter of Shadows Lemma, Diam(Shad(g(o), τ )) .×,s b−kgk and thus
since µ is Ahlfors s-regular,

So


µ Shad(g(o), τ ) .×,s b−skgk ≍×,s b−sρ .
1 .×,s b−sρ #{g ∈ G : ρ − τ ≤ kgk ≤ ρ}

and thus
#{g ∈ G : kgk ≤ ρ} &×,s bsρ .
Since s < δ was arbitrary, we get
lim inf
ρ→∞

logb #{g ∈ G : kgk ≤ ρ}
≥ δ.
ρ

Combining with (8.1.2) completes the proof.



CHAPTER 17

Quasiconformal measures of geometrically finite
groups
In this chapter we investigate the δ-quasiconformal measure or measures associated to a geometrically finite group. Note that since geometrically finite groups
are of compact type (Theorem 12.4.5), Theorem 15.4.6 guarantees the existence
of a δ-quasiconformal measure µ on Λ. However, this measure is not necessarily
unique (Corollary 17.1.8); a sufficient condition for uniqueness is that G is of divergence type (Theorem 1.4.1). In Section 17.1, we generalize a theorem of Dal’bo,
Otal, and Peigne [55, Théorème A] which shows that “most” geometrically finite
groups are of divergence type. In Sections 17.2-17.5 we investigate the geometry of
δ-conformal measures; specifically, in Sections 17.2-17.3 we prove a generalization
of the Global Measure Formula (Theorem 17.2.2), in Sections 17.4 and 17.5 we
investigate the questions of when the δ-conformal measure of a geometrically finite
group is doubling and exact dimensional, respectively.
Standing Assumptions 17.0.3. In this chapter, we assume that
(I) X is regularly geodesic and strongly hyperbolic,
(II) G ≤ Isom(X) is nonelementary and geometrically finite, and δ < ∞.1

Moreover, we fix a complete set of inequivalent parabolic points P ⊆ Λbp , and for

each p ∈ P we write δp = δ(Gp ), and let Sp ⊆ Ep be a p-bounded set satisfying
(A)-(C) of Lemma 12.3.6. Finally, we choose a number t0 > 0 large enough so that
if
Hp = Hp,t0 = {x ∈ X : B p (o, x) > t0 }
H = {g(Hp ) : p ∈ P, g ∈ G},
then the collection H is disjoint (cf. Proof of Theorem 12.4.5(B3) ⇒ (A)).
17.1. Sufficient conditions for divergence type
In the Standard Case, all geometrically finite groups are of divergence type
[165, Proposition 2]; however, once one moves to the more general setting of pinched
Hadamard manifolds, one has examples of geometrically finite groups of convergence
1Note that by Corollary 12.4.17(ii), we have δ < ∞ if and only if δ < ∞ for all p ∈ P .
p
281

282

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

type [55, Théorème C]. On the other hand, Proposition 16.6.5 shows that for every
δ-conformal measure µ, G is of divergence type if and only if µ(Λ \ Λr) = 0. Now by
Theorem 12.4.5, Λ \ Λr = Λbp = G(P ), so the condition µ(Λ \ Λr ) = 0 is equivalent

to the condition µ(P ) = 0. To summarize:

Observation 17.1.1. The following are equivalent:
(A) G is of divergence type.
(B) There exists a δ-conformal measure µ on Λ satisfying µ(P ) = 0.
(C) Every δ-conformal measure µ on Λ satisfies µ(P ) = 0.
(D) There exists a unique δ-conformal measure µ on Λ, and it satisfies µ(P ) =
0.
In particular, every convex-cobounded group is of divergence type.
It is of interest to ask for sufficient conditions which are not phrased in terms
of measures. We have the following:
Theorem 17.1.2 (Cf. [165, Proposition 2], [55, Théorème A]). If δ > δp for
all p ∈ P , then G is of divergence type.
Proof. We will demonstrate (B) of Observation 17.1.1. Let µ be the measure
constructed in the proof of Theorem 15.4.6, fix p ∈ P , and we will show that

µ(p) = 0. In what follows, we use the same notation as in the proof of Theorem
15.4.6. Since G is strongly discrete, we can let ρ be small enough so that Sρ = G(o).
For any neighborhood U of p, we have
(17.1.1)

µ(p) ≤ lim inf µs (U ) = lim inf
sցδ

sցδ

1
Σs,k

X

k(x)e−skxk .

x∈G(o)∩U

Lemma 17.1.3.
hh(o)|xio ≍+ 0 ∀x ∈ Sp .
Proof. Since Sp is p-bounded, Gromov’s inequality implies that
hh(o)|xio ∧ hh(o)|pio ≍+ 0
for all h ∈ Gp and x ∈ Sp . Denote the implied constant by σ. For all h ∈ Gp such
that hh(o)|pio > σ, we have hh(o)|xio ≤ σ ∀x ∈ Sp . Since this applies to all but
finitely many h ∈ Gp , (c) of Proposition 3.3.3 completes the proof.

⊳

Let T be a transversal of Gp \G such that T (o) ⊆ Sp . Then by Lemma 17.1.3,
kh(x)k ≍+ khk + kxk ∀h ∈ Gp ∀x ∈ T (o).

17.1. SUFFICIENT CONDITIONS FOR DIVERGENCE TYPE

Thus for all s > δ and V ⊆ X,
X
k(x)e−skxk =
(17.1.2)

X

X

283

k(ekxk )e−skxk

h∈Gp x∈hT (o)∩U

x∈G(o)∩U

≍×

X

h∈Gp

X

k(ekhk+kxk )e−s[khk+kxk] .

x∈T (o)∩h−1 (U)

Now fix 0 < ε < δ − δp , and note that by (15.4.1),
k(R) ≤ k(λR) .×,ε λε k(R) ∀λ > 1 ∀R ≥ 1.
Thus setting V = U in (17.1.2) gives
X
X
k(x)e−skxk .×,ε

e−(s−ε)khk

h∈Gp
h(Sp )∩U6=∅

x∈G(o)∩U

X

k(x)e−skxk ,

x∈T (o)

while setting V = X gives
X
X
X
k(x)e−skxk .
k(x)e−skxk &×
e−skhk
Σs,k =
h∈Gp

x∈G(o)

x∈T (o)

Dividing these inequalities and combining with (17.1.1) gives
X
X
1
1
µ(p) .×,ε lim inf
e−(s−ε)khk =
sցδ Σs (Gp )
Σδ (Gp )
h∈Gp
h(Sp )∩U6=∅

e−(δ−ε)khk .

h∈Gp
h(Sp )∩U6=∅

Note that the right hand series converges since δ − ε > δp by construction. As the

neighborhood U shrinks, the series converges to zero. This completes the proof. 
Combining Theorem 17.1.2 with Proposition 10.3.10 gives the following immediate corollary:
Corollary 17.1.4. If for all p ∈ P , Gp is of divergence type, then G is of

divergence type.

Thus in some sense divergence type can be “checked locally” just like the properties of finite generation and finite Poincaré exponent (cf. Corollary 12.4.17).
Corollary 17.1.5. Every convex-cobounded group is of divergence type.
Remark 17.1.6. It is somewhat awkward that it seems to be difficult or impossible to prove Theorem 17.1.2 via any of the equivalent conditions of Observation
17.1.1 other than (B). Specifically, the fact that the above argument works for the
measure constructed in Theorem 15.4.6 (the “Patterson–Sullivan measure”) but
not for other δ-conformal measures seems rather asymmetric. However, after some
thought one realizes that it would be impossible for a proof along similar lines to
work for every δ-conformal measure. This is because the above proof shows that

284

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

the Patterson–Sullivan measure µ satisfies
µ(p) = 0 for all p ∈ P satisfying δ > δp ,

(17.1.3)

but there are geometrically finite groups for which (17.1.3) does not hold for all δconformal measures µ. Specifically, one may construct geometrically finite groups
of convergence type (cf. [55, Théorème C]) such that δp < δ for some p ∈ P ;

the following proposition shows that there exists a δ-conformal measure for which
(17.1.3) fails:
Proposition 17.1.7. If G is of convergence type, then for each p ∈ P there
exists a δ-conformal measure supported on G(p).
Proof. Let
µ=

X

[g ′ (p)]δ δg(p) ;

g(p)∈G(p)

clearly µ is a δ-conformal measure, but we may have µ(∂X) = ∞. To prove that
this is not the case, as before we let T be a transversal of Gp \G such that T (o) ⊆ Sp .
Then
X
X
X
[g ′ (p)]δ =
µ(∂X) =
[g ′ (p)]δ ≍×
e−δkgk ≤ Σδ (G) < ∞.
g(p)∈G(p)

g∈T −1

g∈T −1


Proposition 17.1.7 yields the following characterization of when there exists a
unique δ-conformal measure:
Corollary 17.1.8. The following are equivalent:
(A) There exists a unique δ-conformal measure on Λ.
(B) Either G is of divergence type, or #(P ) = 1.
17.2. The global measure formula
In this section and the next, we fix a δ-quasiconformal measure µ, and ask the
following geometrical question: Given η ∈ Λ and r > 0, can we estimate µ(B(η, r))?
If G is convex-cobounded, then we can show that µ is Ahlfors δ-regular (Corollary
17.2.3), but in general the measure µ(B(η, r)) will depend on the point η, in a
manner described by the global measure formula. To describe the global measure
formula, we need to introduce some notation:
Notation 17.2.1. Given ξ = g(p) ∈ Λbp , let tξ > 0 be the unique number such
that
Hξ = Hξ,tξ = g(Hp ) = g(Hp,t0 ),

17.2. THE GLOBAL MEASURE FORMULA

285

b(η, t)

log m(η, t)

Figure 17.2.1. A possible (approximate) graph of the functions
t 7→ b(η, t) and t 7→ log m(η, t) (cf. (17.2.1) and (17.2.6)). The
graph indicates that there are at least two inequivalent parabolic
points p1 , p2 ∈ P , which satisfy Npi (R) ≍× R2δ Ipi (R) ≍× Rki for
some k1 < 2δ < k2 . The dotted line in the second graph is just the
line y = −δt.
Note the relation between the two graphs, which may be either
direct or inverted depending on the functions Np . Specifically, the
relation is direct for the first cusp but inverted for the second cusp.
i.e. tξ = t0 + B ξ (o, g(o)). (Note that tp = t0 for all p ∈ P .) Fix θ > 0 large
to be determined below (cf. Proposition 17.2.5). For each η ∈ Λ and t > 0, let
ηt = [o, η]t , and write

S


e−δt
ηt ∈
/ (H )


(17.2.1) m(η, t) = e−δtξ [Ip (et−tξ −θ ) + µ(p)]
ηt ∈ Hξ and t ≤ hξ|ηio



e−δ(2hξ|ηio −tξ ) N (e2hξ|ηio −t−tξ −θ ) η ∈ H and t > hξ|ηi
p
t
ξ
o

(cf. Figure 17.2.1.) Here we use the notation
X
Ip (R) =
khk−2δ
p
h∈Gp
khkp ≥R

Np (R) = NEp ,Gp (R) = #{h ∈ Gp : khkp ≤ R}
where
khkp = Dp (o, h(o)) = e(1/2)khk ∀h ∈ Gp .

286

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

Theorem 17.2.2 (Global measure formula; cf. [160, Theorem 2] and [153,
Théorème 3.2]). For all η ∈ Λ and t > 0,
m(η, t + σ) .× µ(B(η, e−t )) .× m(η, t − σ),

(17.2.2)

where σ > 0 is independent of η and t (but may depend on θ).
Corollary 17.2.3. If G is convex-cobounded, then
µ(B(η, r)) ≍× rδ ∀η ∈ Λ ∀0 < r ≤ 1,

(17.2.3)

i.e. µ is Ahlfors δ-regular.
Proof. If G is convex-cobounded then H = , so m(η, t) = e−δt ∀η, t, and
thus (17.2.2) reduces to (17.2.3).

Remark 17.2.4. Corollary 17.2.3 can be deduced directly from Lemma 17.3.7
below.
We will prove Theorem 17.2.2 in the next section. For now, we investigate more
closely the function t 7→ m(η, t) defined by (17.2.1). The main result of this section
is the following proposition, which will be used in the proof of Theorem 17.2.2:
Proposition 17.2.5. If θ is chosen sufficiently large, then for all η ∈ Λ and
0 < t1 < t2 ,
(17.2.4)

m(η, t2 ) .×,θ m(η, t1 ).

The proof of Proposition 17.2.5 itself requires several lemmas.
Lemma 17.2.6. Fix ξ, η ∈ ∂X and t > 0, and let x = ηt . Then
B ξ (o, x) ≍+ t ∧ (2hξ|ηio − t).

(17.2.5)

Proof. Since ho|ηix = 0, Gromov’s inequality gives ho|ξix ∧ hξ|ηix ≍+ 0.

Case 1: ho|ξix ≍+ o. In this case, by (h) of Proposition 3.3.3,

B ξ (o, x) = − Bξ (x, o) = −[2ho|ξix − kxk] ≍+ kxk = t,
while (g) of Proposition 3.3.3 gives
1
1
hξ|ηio = hξ|ηix + [B ξ (o, x) + B η (o, x)] &+ [t + t] = t;
2
2
thus B ξ (o, x) ≍+ t ≍+ t ∧ (2hξ|ηio − t).
Case 2: hξ|ηix ≍+ o. In this case, (g) of Proposition 3.3.3 gives
hξ|ηio ≍+

1
1
1
[B ξ (o, x) + B η (o, x)] = [B ξ (o, x) + t] .+ [t + t] = t;
2
2
2

thus B ξ (o, x) ≍+ 2hξ|ηio − t ≍+ t ∧ (2hξ|ηio − t).


17.2. THE GLOBAL MEASURE FORMULA

Corollary 17.2.7. The function

0
(17.2.6)
b(η, t) =
t ∧ (2hξ|ηi − t) − t
o
ξ

ηt ∈
/

287

S
(H )

ηt ∈ Hξ

satisfies

b(η, t + τ ) ≍+,τ b(η, t − τ ).

(17.2.7)

Proof. Indeed, by Lemma 17.2.6,

0
b(η, t) ≍+
B ξ (o, ηt ) − tξ

ηt ∈
/

S
(H )

ηt ∈ Hξ

= 0 ∨ max (B ξ (o, ηt ) − tξ ).
ξ∈Λbp

The right hand side is 1-Lipschitz continuous with respect to t, which demonstrates
(17.2.7).



Lemma 17.2.8. For all ξ ∈ G(p) ⊆ Λbp , p ∈ P , there exists g ∈ G such that
(17.2.8)

ξ = g(p), kgk ≍+ tξ , and {η ∈ ∂X : [o, η] ∩ Hξ 6= } ⊆ Shad(g(o), σ),

where σ > 0 is independent of ξ.
Proof. Write ξ = g(p) for some g ∈ G. Since x := ξtξ ∈ ∂Hξ , Lemma
12.3.6(D) shows that
d(g −1 (x), h(o)) ≍+ 0
for some h ∈ Gp . We claim that gh is the desired isometry. Clearly kghk ≍+ kxk =
tξ . Fix η ∈ ∂X such that [o, η] ∩ Hξ 6= , say ηt ∈ Hξ . By Lemma 17.2.6, we have
kxk = tξ < B ξ (o, ηt ) ≍+ t ∧ (2hξ|ηio − t) ≤ hξ|ηio ≤ hx|ηio ,
i.e. η ∈ Shad(x, σ) ⊆ Shad(g(o), σ + τ ) for some σ, τ > 0.
Proof of Proposition 17.2.5. Fix η ∈ Λ and 0 < t1 < t2 .



Case 1: ηt1 , ηt2 ∈ Hξ for some ξ = g(p) ∈ Λbp , g satisfying (17.2.8). In this case,
(17.2.4) follows immediately from (17.2.1) unless t1 ≤ hξ|ηio < t2 . If the
latter holds, then

lim

m(η, t) = e−δtξ [Ip (ehξ|ηio −tξ −θ ) + µ(p)]

lim

m(η, t) = e−δ(2hξ|ηio −tξ ) Np (ehξ|ηio −tξ −θ ).

m(η, t1 ) ≥

tրhξ|ηio

m(η, t2 ) ≤

tցhξ|ηio

Consequently, to demonstrate (17.2.4) it suffices to show that
(17.2.9)

Np (et ) .×,θ e2δt Ip (et ),
where t := hξ|ηio − tξ − θ > 0.

288

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

To demonstrate (17.2.9), let ζ = g −1 (η) ∈ Λ. We have
hp|ζio = hξ|ηig(o) ≍+ hξ|ηio − kgk ≍+ hξ|ηio − tξ = t + θ
and thus
Dp (o, ζ) ≍× et+θ .
Since p is a bounded parabolic point, there exists hζ ∈ Gp such that

Dp (hζ (o), ζ) .× 1. Denoting all implied constants by C, we have
C −1 et+θ − C ≤ Dp (o, ζ) − Dp (hζ (o), ζ) ≤ khζ kp

≤ Dp (o, ζ) + Dp (hζ (o), ζ) ≤ Cet+θ + C.

Choosing θ ≥ log(4C), we have
2et ≤ khζ kp ≤ 2Cet+θ unless et+θ ≤ 2C 2 .
If 2et ≤ khζ kp ≤ 2Cet+θ , then for all h ∈ Gp satisfying khkp ≤ et we have
et ≤ khζ hkp .×,θ et ; it follows that
X
≍×,θ e−2δt Np (et ),
khζ hk−2δ
Ip (et ) ≥
p
h∈Gp

thus demonstrating (17.2.9). On the other hand, if et+θ ≤ 2C 2 , then both

sides of (17.2.9) are bounded from above and below independent of t.
Case 2: No such ξ exists. In this case, for each i write ηi ∈ Hξi for some ξi =

gi (pi ) ∈ Λbp if such a ξi exists. If ξ1 exists, let s1 > t1 be the smallest
number such that ηs1 ∈ ∂Hξ1 , and if ξ2 exists, let s2 < t2 be the largest
number such that ηs2 ∈ ∂Hξ2 . If ξi does not exist, let si = ti . Then
t1 ≤ s1 ≤ s2 ≤ t2 . Since m(η, si ) = e−δsi , we have m(η, s2 ) ≤ m(η, s1 ),
so to complete the proof it suffices to show that
m(η, s1 ) .×,θ m(η, t1 ) and
m(η, s2 ) &×,θ m(η, t2 ).

By Case 1, it suffices to show that
m(η, s1 ) .× lim m(η, t) if ξ1 exists, and
tրs1

m(η, s2 ) &× lim m(η, t) if ξ2 exists.
tցs2

Comparing with (17.2.1), we see that the desired formulas are
e−δs1 .× e−δ(2hξ|ηio −tξ1 ) Np (e2hξ1 |ηio −s1 −tξ1 )

e−δs2 &× e−δtξ2 [Ip (es2 −tξ2 ) + µ(p)],

17.3. PROOF OF THE GLOBAL MEASURE FORMULA

289

which follow upon observing that the definitions of s1 and s2 imply that
s1 ≍+ 2hξ|ηio − tξ1 and s2 ≍+ tξ2 (cf. Lemma 17.2.6).


17.3. Proof of the global measure formula
Although we have finished the proof of Proposition 17.2.5, we still need a few
lemmas before we can begin the proof of Theorem 17.2.2. Throughout these lemmas, we fix p ∈ P , and let
Rp = sup Dp (o, x) < ∞.
x∈Sp

Here Sp ⊆ Ep is a p-bounded set satisfying Λ \ {p} ⊆ Gp (Sp ), as in Standing
Assumptions 17.0.3.
Lemma 17.3.1. For all A ⊆ Gp ,
!
[
X
X
(17.3.1)
µ
h(Sp ) ≍×
e−δkhk =
khk−2δ
p .
h∈A

h∈A

h∈A

Proof. As the equality follows from Observation 6.2.10, we proceed to demonstrate the asymptotic. By Lemma 17.1.3, there exists σ > 0 such that Sp ⊆
Shadh−1 (o) (o, σ) for all h ∈ Gp . Then by the Bounded Distortion Lemma 4.5.6,
Z
′
(h )δ dµ ≍×,σ e−δkhk µ(Sp ) ≍× e−δkhk .
µ(h(Sp )) =
Sp

(In the last asymptotic, we have used the fact that µ(Sp ) > 0, which follows from
the fact that Λ \ {p} ⊆ Gp (Sp ) together with the fact that µ is not a pointmass
(Corollary 15.4.3).) Combining with the subadditivity of µ gives the . direction of
the first asymptotic of (17.3.1). To get the & direction, we observe that since Sp
is p-bounded, the strong discreteness of Gp implies that Sp ∩ h(Sp ) 6=  for only
finitely many h ∈ Gp ; it follows that the function η 7→ #{h ∈ Gp : η ∈ h(Sp )} is

bounded, and thus
µ

[

h∈A

!

h(Sp )

≍×
=

Z

X

#{h ∈ Gp : η ∈ h(Sp )} dµ(η)
µ(h(Sp ))

h∈A

≍×

X

e−δkhk .

h∈A



290

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

ξ=∞

o

ηt

ξ
η

ηt

Hξ

ηtξ
g(o)

o

η

Figure 17.3.1. Cusp excursion in the ball model (left) and upper
half-space model (right). Since ξ = g(p) ∈ B(η, e−t ), our estimate of µ(B(η, e−t )) is based on the function Ip , which captures
information “at infinity” about the cusp p. In the right-hand picture, the measure of B(η, e−t ) can be estimated by considering the
measure from the perspective of g(o) of a small ball around ξ.
Corollary 17.3.2. For all r > 0,
 
 

2
1
(17.3.2)
Ip
.× µ B(p, r) \ {p} .× Ip
r
2r
Proof. Since
[
h(Sp ) ⊆ B(p, 1/R) \ {p} = Ep \ Bp (o, R) ⊆
h∈Gp
khkp ≥R+Rp

Lemma 17.3.1 gives
Ip



1
+ Rp
r



.× µ(B(p, r)) .× Ip



[

h(Sp ),

h∈Gp
khkp ≥R−Rp


1
− Rp ,
r

thus proving the lemma if r ≤ 1/(2Rp ). But when r > 1/(2Rp ), all terms of (17.3.2)

are bounded from above and below independent of r.



Adding µ(p) to all sides of (17.3.2) gives
 
 
2
1
(17.3.3)
Ip
+ µ(p) .× µ(B(p, r)) .× Ip
+ µ(p).
r
2r
Corollary 17.3.3 (Cf. Figure 17.3.1). Fix η ∈ Λ and t > 0 such that ηt ∈ Hξ

for some ξ = g(p) ∈ Λbp satisfying t ≤ hξ|ηio − log(2). Then

e−δtξ [Ip (et−tξ +σ ) + µ(p)] .× µ B(η, e−t ) .× e−δtξ [Ip (et−tξ −σ ) + µ(p)],
where σ > 0 is independent of η and t.

17.3. PROOF OF THE GLOBAL MEASURE FORMULA

291

Proof. The inequality hξ|ηio ≥ t + log(2) implies that
B(ξ, e−t /2) ⊆ B(η, e−t ) ⊆ B(ξ, 2e−t ).
Without loss of generality suppose that g satisfies (17.2.8). Since t > tξ , (4.5.9)
guarantees that B(ξ, 2e−t ) ⊆ Shad(g(o), σ0 ) for some σ0 > 0 independent of η and
t. Then by the Bounded Distortion Lemma 4.5.6, we have


B p, e−(t−tξ ) /(2C) ⊆ g −1 (B(ξ, e−t /2))
⊆ g −1 (B(η, e−t ))

⊆ g −1 (B(ξ, 2e−t ))


⊆ B p, 2Ce−(t−tξ )

for some C > 0, and thus

 

 
e−δtξ µ B p, e−(t−tξ ) /(2C) .× µ(B(η, e−t )) .× e−δtξ µ B p, 2Ce−(t−tξ ) .

Combining with (17.3.3) completes the proof.



Lemma 17.3.4. For all η ∈ Λ \ {p} and 3Rp ≤ R ≤ Dp (o, η)/2,
Dp (o, η)−2δ Np (R/2) .× µ(Bp (η, R)) .× Dp (o, η)−2δ Np (2R).
Proof. Since η ∈ Λ\{p} ⊆ Gp (Sp ), there exists hη ∈ Gp such that η ∈ hη (Sp ).
Since
[
[
hη h(Sp ) ⊆ Bp (η, R) ⊆
hη h(Sp ),
h∈Gp
khkp ≤R−Rp

Lemma 17.3.1 gives
X

h∈Gp
khkp ≤R−Rp

h∈Gp
khkp ≤R+Rp

khη hk−2δ
.× µ(Bp (η, R)) .×
p

X

h∈Gp
khkp ≤R+Rp

khη hk−2δ
p .

The proof will be complete if we can show that for each h ∈ Gp such that khkp ≤
R + Rp , we have

khη hkp ≍× Dp (o, η).

(17.3.4)
And indeed,

Dp (η, hη h(o)) ≤ Dp (η, hη (o)) + khkp ≤ Rp + (R + Rp ) ≤

5
Dp (o, η),
6

demonstrating (17.3.4) with an implied constant of 6.



Corollary 17.3.5. For all η ∈ Λ \ {p} and 6Rp D(p, η)2 ≤ r ≤ D(p, η)/4, we

have

2δ

D(p, η) Np



r
4D(p, η)2



2δ

.× µ(B(η, r)) .× D(p, η) Np



4r
D(p, η)2



.

292

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS



r
Proof. By (4.2.2), for every ζ ∈ Bp η, D(p,η)(D(p,η)+r)
we have that
D(η, ζ) =

Dp (η, ζ)
Dp (η, ζ)
≤
Dp (o, η)Dp (o, ζ)
Dp (o, η)(Dp (o, η) − Dp (η, ζ))
≤



r
D(p,η)(D(p,η)+r)

Dp (o, η) Dp (o, η) −

= r.

r
D(p,η)(D(p,η)+r)



Analogously, (4.2.2) also implies that for every ζ ∈ B(η, r) we have
D(η, ζ)
D(p, η)D(p, ζ)
r
≤
·
D(p, η) (D(p, η) − r)

Dp (η, ζ) =

Combining these inequalities gives us that




r
r
⊆ B(η, r) ⊆ Bp η,
Bp η,
D(p, η)(D(p, η) + r)
D(p, η)(D(p, η) − r)
Now since r ≤ D(p, η)/4, we have




2r
r
⊆ B(η, r) ⊆ Bp η,
.
Bp η,
2D(p, η)2
D(p, η)2
On the other hand, since 6Rp D(p, η)2 ≤ r ≤ D(p, η)/4, we have
3Rp ≤

r
2r
D(p, η)
≤
≤
2
2
2D(p, η)
D(p, η)
2

whereupon Lemma 17.3.4 completes the proof.



Corollary 17.3.6 (Cf. Figure 17.3.2). Fix η ∈ Λ and t > 0 such that ηt ∈ Hξ
for some ξ = g(p) ∈ Λbp . If
hξ|ηio + τ ≤ t ≤ 2hξ|ηio − tξ − τ,

(17.3.5)
then
(17.3.6)


e−δ(2hξ|ηio −tξ ) Np (e2hξ|ηio −tξ −t−σ ) .× µ B(η, e−t )

.× e−δ(2hξ|ηio −tξ ) Np (e2hξ|ηio −tξ −t+σ )

where σ, τ > 0 are independent of η and t.
Proof. Without loss of generality suppose that g satisfies (17.2.8), and write
ζ = g −1 (η). Since t > tξ , (4.5.9) guarantees that B(η, e−t ) ⊆ Shad(g(o), σ0 ) for
some σ0 > 0 independent of η and t. Then by the Bounded Distortion Lemma
4.5.6, we have
B(ζ, e−(t−tξ ) /C) ⊆ g −1 (B(η, e−t )) ⊆ B(ζ, Ce−(t−tξ ) )

17.3. PROOF OF THE GLOBAL MEASURE FORMULA

293

ξ=∞

ξ

o

ηt

η

Hξ

ηtξ

ηt
g(o)

o

η

Figure 17.3.2. Cusp excursions in the ball model (left) and upper
half-space model (right). Since ξ = g(p) ∈
/ B(η, e−t ), our estimate
of µ(B(η, e−t )) is based on the function Np , which captures “local” information about the cusp p. In the right-hand picture, the
measure of B(η, e−t ) can be estimated by considering the measure
from the perspective of g(o) of a large ball around η taken with
respect to the Dξ -metametric.

for some C > 0, and thus
e−δtξ µ(B(ζ, e−(t−tξ ) /C)) .× µ(B(η, e−t )) .× e−δtξ µ(B(ζ, Ce−(t−tξ ) )).
If
D(p, ζ)
e−(t−tξ )
≤ Ce−(t−tξ ) ≤
,
C
4
then Corollary 17.3.5 guarantees that
 −(t−tξ ) 
e
e−δtξ D(p, ζ)2δ Np
.× µ(B(η, e−t ))
4CD(p, ζ)2


4Ce−(t−tξ )
−δtξ
2δ
.
.× e
D(p, ζ) Np
D(p, ζ)2

(17.3.7)

6Rp D(p, η)2 ≤

On the other hand, since ξ, η ∈ Shad(g(o), σ0 ), the Bounded Distortion Lemma
4.5.6 guarantees that
D(p, ζ) ≍× etξ D(ξ, η) = e−(hξ|ηio −tξ ) .
Denoting the implied constant by K, we deduce (17.3.6) with σ = log(4CK 2 ). The
proof is completed upon observing that if τ = log(4CK ∨ 6Rp CK 2 ), then (17.3.5)
implies (17.3.7).


294

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

Lemma 17.3.7 (Cf. Lemma 15.4.1). Fix η ∈ Λ and t > 0 such that ηt ∈
/

Then

µ(B(η, e−t )) ≍× e−δt .

S
(H ).

Proof. By (12.4.2), there exists g ∈ G such that d(g(o), ηt ) ≍+ 0. By (4.5.9),
we have B(η, e−t ) ⊆ Shad(g(o), σ) for some σ > 0 independent of η, t. It follows

that

µ(B(η, e−t )) ≍× e−δt µ(g −1 (B(η, e−t ))).

To complete the proof it suffices to show that µ(g −1 (B(η, e−t ))) is bounded from
below. By the Bounded Distortion Lemma 4.5.6,
g −1 (B(η, e−t )) ⊇ B(g −1 (η), ε)
for some ε > 0 independent of η, t. Now since G is of compact type, we have
inf µ(B(x, ε)) ≥ min µ(B(x, ε/2)) > 0

x∈Λ

x∈Sε/2

where Sε/2 is a maximal ε/2-separated subset of Λ. This completes the proof.



We are now ready to prove Theorem 17.2.2:
Proof of Theorem 17.2.2. Let σ0 denote the implied constant of (17.2.5).
Then by (17.2.1), for all η ∈ Λ, t > 0, and ξ ∈ Λbp ,
(17.3.8)



e−δtξ [Ip (et−tξ −θ ) + µ(p)]
tξ + σ0 ≤ t ≤ hξ|ηio


−δ(2hξ|ηi
−t
)
2hξ|ηi
−t−t
−θ
o
o
ξ
ξ
m(η, t) = e
Np (e
) hξ|ηio < t ≤ 2hξ|ηio − tξ − σ0



unknown
otherwise
Applying this formula to Corollaries 17.3.3 and 17.3.6 yields the following:
Lemma 17.3.8. There exists τ ≥ σ0 such that for all η ∈ Λ and t > 0.
(i) If for some ξ, tξ + τ ≤ t ≤ hξ|ηio − τ , then (17.2.2) holds.

(ii) If for some ξ, hξ|ηio + τ ≤ t ≤ 2hξ|ηio − tξ − τ , then (17.2.2) holds.
Now fix η ∈ Λ, and let
n
o [
[
[
[hξ|ηio +τ, 2hξ|ηio −tξ −τ ].
[tξ +τ, hξ|ηio −τ ]∪
A = t > 0 : ηt ∈
/ (H ) ∪
ξ∈Λbp

ξ∈Λbp

Then by Lemmas 17.3.7 and 17.3.8, (17.2.2)σ=τ holds for all t ∈ A.
Claim 17.3.9. Every interval of length 2τ intersects A.
Proof. If [s − τ, s + τ ] does not intersect A, then by connectedness, there
exists ξ ∈ Λbp such that ηt ∈ Hξ for all t ∈ [s − τ, s + τ ]. By Lemma 17.2.6,
the fact that ηs±τ ∈ Hξ implies that tξ ≤ s ≤ 2hξ|ηio − tξ (since τ ≥ σ0 ). If

17.4. GROUPS FOR WHICH µ IS DOUBLING

295

s ≤ hξ|ηio , then [s − τ, s + τ ] ∩ [tξ + τ, hξ|ηio − τ ] 6= , while if s ≥ hξ|ηio , then
[s − τ, s + τ ] ∩ [hξ|ηio + τ, 2hξ|ηio − tξ − τ ] 6= .

⊳

Thus for all t > 0, there exist t± ∈ A such that t − 2τ ≤ t− ≤ t ≤ t+ ≤ t − 2τ ;
then

m(η, t + 3τ ) .× m(η, t+ + τ ) .× µ B(η, e−t+ )

≤ µ B(η, e−t )

≤ µ B(η, e−t− ) .× m(η, t− − τ ) .× m(η, t − 3τ ),

i.e. (17.2.2)σ=3τ holds.



17.4. Groups for which µ is doubling
Recall that a measure µ is said to be doubling if for all η ∈ Supp(µ) and r > 0,
µ(B(η, 2r)) ≍× µ(B(η, r)). In the Standard Case, the Global Measure Formula

implies that the δ-conformal measure of a geometrically finite group is always doubling (Example 17.4.11). However, in general there are geometrically finite groups
whose δ-conformal measures are not doubling (Example 17.4.12). It is therefore of
interest to determine necessary and sufficient conditions on a geometrically finite
group for its δ-conformal measure to be doubling. The Global Measure Formula
immediately yields the following criterion:
Lemma 17.4.1. µ is doubling if and only if the function m satisfies
(17.4.1)

m(η, t + τ ) ≍×,τ m(η, t − τ ) ∀η ∈ Λ ∀t, τ > 0.

Proof. If (17.4.1) holds, then (17.2.2) reduces to
(17.4.2)

µ(B(η, e−t )) ≍× m(η, t),

and then (17.4.1) shows that µ is doubling. On the other hand, if µ is doubling,
then (17.2.2) implies that
m(η, t − τ ) .× µ(B(η, e−(t−τ −σ) )) ≍× µ(B(η, e−(t+τ +σ) )) .× m(η, t + τ );
combining with Proposition 17.2.5 shows that (17.4.1) holds.



Of course, the criterion (17.4.1) is not very useful by itself, since it refers to the
complicated function m. In what follows we find more elementary necessary and
sufficient conditions for doubling. First we must introduce some terminology.
Definition 17.4.2. A function f : [1, ∞) → [1, ∞) is called doubling if there

exists β > 1 such that
(17.4.3)

f (βR) .×,β f (R) ∀R ≥ 1,

296

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

and codoubling if there exists β > 1 such that
f (βR) − f (R) &×,β f (R) ∀R ≥ 1.

(17.4.4)

Observation 17.4.3. If there exists β > 1 such that
Np (βR) > Np (R) ∀R ≥ 1,
then Np is codoubling.
Proof. Fix R ≥ 1; there exists h ∈ Gp such that 2R < khkp ≤ 2βR. We have
h{j ∈ Gp : kjkp ≤ R} ⊆ {j ∈ Gp : R < kjkp ≤ (2β + 1)R},
and taking cardinalities gives

Np (R) ≤ Np (2β + 1)R − Np (R).



We are now ready to state a more elementary characterization of when µ is
doubling:
Proposition 17.4.4. µ is doubling if and only if all of the following hold:
(I) For all p ∈ P , Np is both doubling and codoubling.

(II) For all p ∈ P and R ≥ 1,
(17.4.5)

Ip (R) ≍× R−2δ Np (R).

(III) G is of divergence type.
Moreover, (II) can be replaced by
(II′ ) For all p ∈ P and R ≥ 1,
(17.4.6)

Iep (R) :=

∞
X

k=0

e−2δk Np (ek R) ≍× Np (R).

Proof that (I)-(III) imply µ doubling. Fix η ∈ Λ and t, τ > 0, and we will
demonstrate (17.4.1). By (II), (III), and Observation 17.1.1, we have

S


e−δt
ηt ∈
/ (H )


m(η, t) ≍× e−δtξ e−2δ(t−tξ −θ) Np (et−tξ −θ )
ηt ∈ Hξ and t ≤ hξ|ηio



e−δ(2hξ|ηio −tξ ) N (e2hξ|ηio −t−tξ −θ ) η ∈ H and t > hξ|ηi
(17.4.7)
p
t
ξ
o

S
1
ηt ∈
/ (H )
≍× e−δt
e−δb(η,t) N (eb(η,t)−θ ) η ∈ H
p

t

g(p)

where b(η, t) is as in (17.2.6). Let t± = t ± τ . We split into two cases:

17.4. GROUPS FOR WHICH µ IS DOUBLING

297

Case 1: ηt+ , ηt− ∈ Hg(p) for some g(p) ∈ Λbp . In this case, (17.4.1) follows from

Corollary 17.2.7 together with the fact that Np is doubling.
S
Case 2: ηt+s ∈
/ (H ) for some s ∈ [−τ, τ ]. In this case, Corollary 17.2.7 shows
that b(η, t± ) ≍+,τ 0 and thus

m(η, t+ ) ≍×,τ e−δt ≍×,τ m(η, t− ).

Before continuing the proof of Proposition 17.4.4, we observe that
Ip (R) + R−2δ Np (R) ≍×

X

h∈Gp

(R ∨ khkp )−2δ ≍×
=

∞
X X

h∈Gp k=1
∞
X

k=1

(ek R)−2δ [ek R ≥ khkp ]

(ek R)−2δ Np (ek R)

= R−2δ Iep (R).

In particular, it follows that (17.4.6) is equivalent to

Ip (R) .× R−2δ Np (R).

(17.4.8)

Proof that (I) and (II′ ) imply (II). Since Np is codoubling, let β > 1 be as

in (17.4.4). Then
X
Ip (R) ≥

h∈Gp
R<khkp ≤βR

(βR)−2δ = (βR)−2δ (Np (βR) − Np (R)) &×,β R−2δ Np (R).

Combining with (17.4.8) completes the proof.



Proof that µ doubling implies (I)-(III) and (II′ ). Since a doubling measure whose topological support is a perfect set cannot have an atomic part, we must
have µ(P ) = 0 and thus by Observation 17.1.1, (III) holds. Since
m(p, t) ≍×,p Ip (et−t0 −θ ) + µ(p) = Ip (et−t0 −θ )
for all sufficiently large t, setting η = p in (17.4.1) shows that the function Ip is
doubling.
Fix η ∈ Λ \ {p}. Let σ0 > 0 denote the implied constant of (17.2.5). For
s ∈ [t0 + σ0 + τ, hp|ηio − τ ], plugging t = 2hp|ηio − s into (17.4.1) and simplifying
using (17.3.8)ξ=p shows that

(17.4.9)

Np (es−τ −t0 −θ ) ≍×,τ Np (es+τ −t0 −θ ).

Since hp|ηio can be made arbitrarily large, (17.4.9) holds for all s ≥ t0 + σ0 + τ . It
follows that Np is doubling.

298

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

Next, we compare the values of m(η, hp|ηio ± τ ). This gives (assuming hp|ηio >

t0 + σ0 + τ )

e−δt0 Ip (ehp|ηio −τ −t0 −θ ) ≍× e−δ(2hp|ηio −t0 ) Np (ehp|ηio −τ −t0 −θ ).
Letting Rη = exp(hp|ηio − τ − t0 − θ), we have
Ip (Rη ) ≍× Rη−2δ Np (Rη ).

(17.4.10)

Now fix ζ ∈ Λ \ {p} and h ∈ Gp , and let η = h(ζ). Then Dp (h(o), η) ≍+,ζ 0, and
thus the triangle inequality gives
1 ≤ Dp (o, η) ≍+,ζ khkp ≥ 1,
and so Rη ≍× Dp (o, η) ≍×,ζ khkp . Combining with (17.4.10) and the fact that the
functions Ip and Np are doubling, we have

Ip (khkp ) ≍× khk−2δ
p Np (khkp )

(17.4.11)

for all h ∈ Gp .
Now fix 1 ≤ R1 < R2 such that khi kp = Ri for some h1 , h2 ∈ Gp , but such that
the formula R1 < khkp < R2 is not satisfied for any h ∈ Gp . Then

lim Ip (R) = lim Ip (R) and lim Np (R) = lim Np (R).

RցR1

RրR2

RցR1

RրR2

On the other hand, applying (17.4.11) with h = h1 , h2 gives
Ip (Ri ) ≍× Ri−2δ Np (Ri ).
Since Ip and Np are doubling, we have
R1−2δ ≍×

limRցR1 Ip (R)
limRրR2 Ip (R)
Ip (R1 )
=
≍×
Np (R1 )
limRցR1 Np (R)
limRրR2 Np (R)
≍×

Ip (R2 )
≍× R2−2δ
Np (R2 )

and thus R1 ≍× R2 . Since R1 , R2 were arbitrary, Observation 17.4.3 shows that

Np is codoubling. This completes the proof of (I).

It remains to demonstrate (II) and (II′ ). Given any R ≥ 1, since Np is codoubling, we may find h ∈ Gp such that khkp ≍× R; combining with (17.4.11) and the

fact that Ip and Np are doubling gives (17.4.5) and (17.4.8), demonstrating (II)
and (II′ ).

We note that the proof actually shows the following (cf. (17.4.7)):
Corollary 17.4.5. If µ is doubling, then

1
µ(B(η, e−t )) ≍× e−δt
e−δb(η,t) N (eb(η,t) )
p

ηt ∈
/

S

(H )

ηt ∈ Hg(p)

17.4. GROUPS FOR WHICH µ IS DOUBLING

299

for all η ∈ Λ, t > 0. Here b(η, t) is as in (17.2.6).
Although Proposition 17.4.4 is the best necessary and sufficient condition we
can give for doubling, in what follows we give necessary conditions and sufficient
conditions which are more elementary (Proposition 17.4.8), although the necessary
conditions are not the same as the sufficient conditions. In practice these conditions
are usually powerful enough to determine whether any given measure is doubling.
To state the result, we need the concept of the polynomial growth rate of a
function:
Definition 17.4.6 (Cf. (11.2.4)). The (polynomial) growth rate of a function
f : [1, ∞) → [1, ∞) is the limit
α(f ) :=

lim

λ,R→∞

log f (λR) − log f (R)
log(λ)

if it exists. If the limit does not exist, then the numbers
α∗ (f ) := lim sup
λ,R→∞

log f (λR) − log f (R)
log(λ)

log f (λR) − log f (R)
λ,R→∞
log(λ)

α∗ (f ) := lim inf

are the upper and lower polynomial growth rates of f , respectively.
Lemma 17.4.7. Let f : [1, ∞) → [1, ∞).
(i) f is doubling if and only if α∗ (f ) < ∞.

(ii) f is codoubling if and only if α∗ (f ) > 0.
(iii)
log f (λ)
log f (λ)
α∗ (f ) ≤ lim inf
≤ lim sup
≤ α∗ (f ).
λ→∞ log(λ)
log(λ)
λ→∞
In particular, α∗ (Np ) ≤ 2δp ≤ α∗ (Np ).
Proof of (i). Suppose that f is doubling, and let C > 1 denote the implied
constant of (17.4.3). Iterating gives
f (β n R) ≤ C n f (R) ∀n ∈ N ∀R ≥ 1
and thus
f (λR) .× λlogβ (C) f (R) ∀λ > 1 ∀R ≥ 1.

It follows that α∗ (f ) ≤ logβ (C) < ∞. The converse direction is trivial.



Proof of (ii). Suppose that f is codoubling, and let C > 1 denote the implied
constant of (17.4.4). Then
f (βR) ≥ (1 + C −1 )f (R) ∀R ≥ 1.

300

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

Iterating gives
f (β n R) ≥ (1 + C −1 )n f (R) ∀n ∈ N ∀R ≥ 1
and thus
f (λR) &× λlogβ (1+C

−1

)

f (R) ∀λ > 1 ∀R ≥ 1.

It follows that α∗ (f ) ≥ logβ (1 + C −1 ) > 0. The converse direction is trivial.



Proof of (iii). Let Rn → ∞. For each n ∈ N,
lim sup
λ→∞

log f (λRn ) − log f (Rn )
log f (λ)
= s := lim sup
·
log(λ)
log(λ)
λ→∞

Thus given s < s, we may find a large number λn > 1 such that
log f (λn Rn ) − log f (Rn )
> s.
log(λn )
Since λn , Rn → ∞ as n → ∞, it follows that α∗ (f ) ≥ s; since s was arbitrary,
α∗ (f ) ≥ s. A similar argument shows that α∗ (f ) ≤ s.
Finally, when f = Np , the equality s = s = 2δp is a consequence of (8.1.2) and
Observation 6.2.10.

We can now state our final result regarding criteria for doubling:
Proposition 17.4.8. In the following list, (A) ⇒ (B) ⇒ (C):
(A) For all p ∈ P , 0 < α∗ (Np ) ≤ α∗ (Np ) < 2δ.
(B) µ is doubling.
(C) For all p ∈ P , 0 < α∗ (Np ) ≤ α∗ (Np ) ≤ 2δ.

Proof of (A) ⇒ (B). Suppose that (A) holds. Then by Lemma 17.4.7, (I) of

Proposition 17.4.4 holds. Since δp ≤ α∗ (Np )/2 < δ for all p ∈ P , Theorem 17.1.2
implies that (III) of Proposition 17.4.4 holds. To complete the proof, we need to
show that (II′ ) of Proposition 17.4.4 holds. Fix s ∈ (α∗ (Np ), 2δ). Since s > α∗ (Np ),
we have
Np (λR) .×,s λs Np (R) ∀λ > 1, R ≥ 1
and thus
Np (R) ≤ Iep (R) .×

∞
X

k=0

e−2δk esk Np (R) ≍× Np (R),

demonstrating (17.4.6) and completing the proof.



Proof of (B) ⇒ (C). Suppose µ is doubling. By (I) of Proposition 17.4.4,

α∗ (Np ) > 0. On the other hand, by (17.4.6) we have

λ−2δ Np (λR) .× Np (R) ∀λ > 1, R ≥ 1
and thus α∗ (Np ) ≤ 2δ.



17.4. GROUPS FOR WHICH µ IS DOUBLING

301

Proposition 17.4.4 shows that if G is a geometrically finite group with δconformal measure µ, then the question of whether µ is doubling is determined
entirely by its parabolic subgroups (Gp )p∈P and its Poincaré set ∆G . A natural
question is when the second input can be removed, that is: if we are told what
the parabolic subgroups (Gp )p∈P are, can we sometimes determine whether µ is
doubling without looking at ∆G ? A trivial example is that if α∗ (Np ) = 0 or
α∗ (Np ) = ∞ for some p ∈ P , then we automatically know that µ is not doubling.
Conversely, the following definition and proposition describe when we can deduce
that µ is doubling:

Definition 17.4.9. A parabolic group H ≤ Isom(X) with global fixed point
p ∈ ∂X is pre-doubling if
(17.4.12)

0 < α∗ (NEp ,H ) ≤ α∗ (NEp ,H ) = 2δH < ∞

and H is of divergence type.
Proposition 17.4.10.
(i) If Gp is pre-doubling for every p ∈ P , then µ is doubling.

(ii) Let H ≤ Isom(X) be a parabolic subgroup, and let g ∈ Isom(X) be a loxodromic isometry such that hg, Hi is a strongly separated Schottky product.
Then the following are equivalent:
(A) H is pre-doubling.

(B) For every n ∈ N, the δn -quasiconformal measure µn of Gn = hg n , Hi
is doubling. Here we assume that δn := δ(Gn ) < ∞.

Proof of (i). For all p ∈ P , the fact that Gp is of divergence type implies
that δ > δp (Proposition 10.3.10); combining with (17.4.12) gives 0 < α∗ (Np ) ≤

α∗ (Np ) < 2δ. Proposition 17.4.8 completes the proof.



Proof of (ii). Since (up to equivalence) the only parabolic point of Gn is the
global fixed point of H (Proposition 12.4.19), the implication (A) ⇒ (B) follows

from part (i). Conversely, suppose that (B) holds. Then by Proposition 17.4.8, we
have
0 < α∗ (NEp ,H ) ≤ α∗ (NEp ,H ) ≤ 2δn < ∞.
Since δn → δH as n → ∞ (Proposition 10.3.7(iv)), taking the limit and combining
with the inequality 2δH ≤ α∗ (NEp ,H ) yields (17.4.12). On the other hand, by

Proposition 17.4.4, for each n, Gn is of divergence type, so applying Proposition
10.3.7(iv) again, we see that H is of divergence type.

Example 17.4.11. If
(17.4.13)

Np (R) ≍× R2δp ∀p ∈ P,

302

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

then the groups (Gp )p∈P are pre-doubling, and thus by Proposition 17.4.10(i), µ is
doubling. Combining with Corollary 17.4.5 gives

1
µ(B(η, e−t )) ≍× e−δt
e(2δξ −δ)b(η,t)

ηt ∈
/

S
(H )

ηt ∈ Hξ

.

This generalizes B. Schapira’s global measure formula [153, Théorème 3.2] to the
setting of regularly geodesic strongly hyperbolic metric spaces.
We remark that the asymptotic (17.4.13) is satisfied whenever X is a finitedimensional algebraic hyperbolic space; see e.g. [137, Lemma 3.5]. In particular,
specializing Schapira’s global measure formula to the settings of finite-dimensional
algebraic hyperbolic spacess and finite-dimensional real hyperbolic spaces give the
global measure formulas of Newberger [137, Main Theorem] and Stratmann–Velani–
Sullivan [160, Theorem 2], [165, Theorem on p.271], respectively.
By contrast, when X = H = H∞ , the asymptotic (17.4.13) is usually not
satisfied. Let us summarize the various behaviors that we have seen for the orbital
counting functions of parabolic groups acting on H∞ , and their implications for
doubling:
Examples 17.4.12 (Examples of doubling and non-doubling Patterson–Sullivan
measures of geometrically finite subgroups of Isom(H∞ )).
1. In the proof of Theorem 11.2.11 (cf. Remark 11.2.12), we saw that if Γ is
a finitely generated virtually nilpotent group and if f : [1, ∞) → [1, ∞) is
a function satisfying

αΓ < α∗ (f ) ≤ α∗ (f ) < ∞,
then there exists a parabolic group H ≤ Isom(H∞ ) isomorphic to Γ whose
orbital counting function is asymptotic to f . Now, a group H constructed
in this way may or may not be pre-doubling; it depends on the chosen
function f . We note that by applying Proposition 17.4.10(ii) to such a
group, one can construct examples of geometrically finite subgroups of
Isom(H∞ ) whose Patterson–Sullivan measures are not doubling. On the
other hand, for any parabolic group H constructed in this way, if H is embedded into a geometrically finite group G with sufficiently large Poincaré
exponent (namely 2δG > α∗ (f )), then the Patterson–Sullivan measure of
G may be doubling (assuming that no other parabolic subgroups of G are
causing problems).
2. In Theorem 14.1.5, we showed that if f : [0, ∞) → N satisfies the condition
∀0 ≤ R1 ≤ R2 f (R1 ) divides f (R2 ),

17.5. EXACT DIMENSIONALITY OF µ

303

then there exists a parabolic subgroup of Isom(H∞ ) whose orbital counting
function is equal to f . This provides even more examples of parabolic
groups which are not pre-doubling. In particular, it provides examples of
parabolic groups H which satisfy either α∗ (NH ) = 0 or α∗ (NH ) = ∞ (cf.
Example 11.2.18); such groups cannot be embedded into any geometrically
finite group with a doubling Patterson–Sullivan measure.
Note that example 2 can be used to construct a geometrically finite group
acting isometrically on an R-tree which does not have a doubling Patterson–Sullivan
measure. On the other hand, example 1 has no analogue in R-trees by Remark 6.1.8.
17.5. Exact dimensionality of µ
We now turn to the question of the fractal dimensions of the measure µ. We
recall that the Hausdorff dimension and packing dimension of a measure µ on ∂X
are defined by the formulas
dimH (µ) = inf {dimH (A) : µ(∂X \ A) = 0}
dimP (µ) = inf {dimP (A) : µ(∂X \ A) = 0} .
If G is of convergence type, then µ is atomic, so dimH (µ) = dimP (µ) = 0. Consequently, for the remainder of this chapter we make the
Standing Assumption 17.5.1. G is of divergence type.
Given this assumption, it is natural to expect that dimH (µ) = dimP (µ) = δ.
Indeed, the inequality dimH (µ) ≤ δ follows immediately from Theorems 1.2.1 and

12.4.5, and in the Standard Case equality holds [160, Proposiiton 4.10]. Even
stronger than the equalities dimH (µ) = dimP (µ) = δ, it is natural to expect that
µ is exact dimensional :
Definition 17.5.2. A measure µ on a metric space (Z, D) is called exact dimensional of dimension s if the limit
(17.5.1)

dµ (η) := lim

t→∞

1
1
log
t
µ(B(η, e−t ))

exists and equals s for µ-a.e. η ∈ Z.
For example, every Ahlfors s-regular measure is exact dimensional of dimension
s.
If the limit in (17.5.1) does not exist, then we denote the lim inf by dµ (η) and
the lim sup by dµ (η).

304

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

Proposition 17.5.3 ([127, §8]). For any measure µ on a metric space (Z, D),
dimH (µ) = ess sup dµ (η)
η∈Z

dimP (µ) = ess sup dµ (η).
η∈Z

In particular, if µ is exact dimensional of dimension s, then
dimH (µ) = dimP (µ) = s.
Combining Proposition 17.5.3 with Lemma 17.3.7 and Observation 17.1.1 immediately yields the following:
Observation 17.5.4. If µ is the Patterson–Sullivan measure of a geometrically
finite group of divergence type, then
dimH (µ) ≤ δ ≤ dimP (µ).
In particular, if µ is exact dimensional, then µ is exact dimensional of dimension δ.
It turns out that µ is not necessarily exact dimensional (Example 17.5.14), but
counterexamples to exact dimensionality must fall within a very narrow window
(Theorem 17.5.9), and in particular if µ is doubling then µ is exact dimensional
(Corollary 17.5.12). As a first step towards these results, we will show that exact
dimensionality is equivalent to a certain Diophantine condition. For this, we need
to recall some results from [73].
17.5.1. Diophantine approximation on Λ. Classically, Diophantine approximation is concerned with the approximation of a point x ∈ R \ Q by a rational
number p/q ∈ Q. The two important quantities are the error term |x − p/q| and

the height q. Given a function Ψ : N → [0, ∞), the point x ∈ R \ Q is said to be
Ψ-approximable if
x−

p
≤ Ψ(q) for infinitely many p/q ∈ Q.
q

In the setting of a group acting on a hyperbolic metric space, we can instead
talk about dynamical Diophantine approximation, which is concerned with the
approximation of a point η ∈ Λ by points g(ξ) ∈ G(ξ), where ξ ∈ Λ is a distinguished
point. For this to make sense, one needs a new definition of error and height: the
error term is defined to be D(g(ξ), η), and the height is defined to be bkgk . (If there
is more than one possibility for g, it may be chosen so as to minimize the height.)
Some motivation for these definitions comes from considering classical Diophantine
approximation as a special case of dynamical Diophantine approximation which
occurs when X = H2 and G = SL2 (Z); see e.g. [73, Observation 1.15] for more
details. Given a function Φ : [0, ∞) → (0, ∞), the point η ∈ Λ is said to be Φ, ξ-well

17.5. EXACT DIMENSIONALITY OF µ

305

approximable if for every K > 0 there exists g ∈ G such that
D(g(ξ), η) ≤ Φ(Kbkgk ) for infinitely many g ∈ G
(cf. [73, Definition 1.36]). Moreover, η is said to be ξ-very well approximable if
ωξ (η) := lim sup
g∈G
g(ξ)→η

− logb D(g(ξ), η)
>1
kgk

(cf. [73, p.9]). The set of Φ, ξ-well approximable points is denoted WAΦ,ξ , while
the set of ξ-very well approximable points is denoted VWAξ . Finally, a point η is
said to be Liouville if ωξ (η) = ∞; the set of Liouville points is denoted Liouvilleξ .
In the following theorems, we return to the setting of Standing Assumptions
17.0.3 and 17.5.1.

Theorem 17.5.5 (Corollary of [73, Theorem 8.1]). Fix p ∈ P , and let Φ :
[0, ∞) → (0, ∞) be a function such that the function t 7→ tΦ(t) is nonincreasing.
Then

(i) µ(WAΦ,p ) = 0 or 1 according to whether the series


X
1
−δkgk
(17.5.2)
e
Ip
ekgk Φ(Kekgk )
g∈G
converges for some K > 0 or diverges for all K > 0, respectively.
(ii) µ(VWAp ) = 0 or 1 according to whether the series
X
(17.5.3)
Σdiv (p, κ) :=
e−δkgk Ip (eκkgk )
g∈G

converges for all κ > 0 or diverges for some κ > 0, respectively.
(iii) µ(Liouvillep ) = 0 or 1 according to whether the series Σdiv (p, κ) converges
for some κ > 0 or diverges for all κ > 0, respectively.

Proof. Standing Assumption 17.5.1, Theorem 1.4.1, and Observation 17.1.1
imply that µ is ergodic and that µ(p) = 0, thus verifying the hypotheses of [73,
Theorem 8.1]. Theorem 17.2.2 shows that
Ip (C1 /r) .×,p µ(B(p, r)) .×,p Ip (C2 /r)

306

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

for some constants C1 ≥ 1 ≥ C2 > 0. Thus for all K > 0,


X
1
−δkgk
e
Ip
ekgk Φ(KC1 ekgk )
g∈G


X
C1
≤
e−δkgk Ip
ekgk Φ(Kekgk )
g∈G
.× [73, (8.1)]

X
.×
e−δkgk Ip
g∈G

≤

X

g∈G

e

−δkgk

Ip



C2
ekgk Φ(Kekgk )



1
ekgk Φ((K/C1 )ekgk )



.

Thus, [73, (8.1)] diverges for all K > 0 if and only if (17.5.2) diverges for all
K > 0. This completes the proof of (i). To demonstrate (ii) and (iii), simply note
T
S
that VWAp = c>0 WAΦc ,p and Liouvillep = c>0 WAΦc ,p , where Φc (t) = t−(1+c) ,
and apply (i). The constant K may be absorbed by a slight change of κ.


Theorem 17.5.6 (Corollary of [73, Theorem 7.1]). For all ξ ∈ Λ and c > 0,
dimH (WAΦc ,ξ ) ≤

δ
,
1+c

where Φc (t) = t−(1+c) as above. In particular, dimH (Liouvilleξ ) = 0, and VWAξ
can be written as the countable union of sets of Hausdorff dimension strictly less
than δ.

(No proof is needed as this follows directly from [73, Theorem 7.1].)
There is a relation between dynamical Diophantine approximation by the orbits
of parabolic points and the lengths of cusp excursions along geodesics. A well-known
example is that a point η ∈ Λ is dynamically badly approximable with respect to

every parabolic point if and only if the geodesic [o, η] has bounded cusp excursion
lengths [73, Proposition 1.21]. The following observation is in a similar vein:

Observation 17.5.7. For η ∈ Λ, we have:
[
b(η, t)
hξ|ηi − tξ
> 0 ⇔ lim sup
>0
η∈
VWAp ⇔ lim sup
tξ
t
t→∞
ξ∈Λbp
p∈P

η∈

[

p∈P

tξ →∞

Liouvillep ⇔ lim sup
ξ∈Λbp
tξ →∞

b(η, t)
hξ|ηi − tξ
= ∞ ⇔ lim sup
= 1.
tξ
t
t→∞

17.5. EXACT DIMENSIONALITY OF µ

307

Proof. If ξ = g(p) ∈ Λbp , then kgk &+ tξ , with ≍+ for at least one value of g

(Lemma 17.2.8). Thus

max ωp (η) = max lim sup
p∈P

p∈P

g∈G
g(p)→η

hξ|ηi
log D(g(p), η)
,
= lim sup
kgk
tξ
ξ∈Λbp
ξ→η

so
(17.5.4)

lim sup
ξ∈Λbp
tξ →∞

hξ|ηi − tξ
= max ωp (η) − 1.
p∈P
tξ

On the other hand, it is readily verified that if [o, η] intersects Hξ , then the function
f (t) = b(η, t)/t attains its maximum at t = hξ|ηio , at which f (t) = hξ|ηio −tξ . Thus
we have that

lim sup
t→∞

(17.5.5)

b(η, t)
hξ|ηio − tξ
b(η, t)
= lim sup sup
= lim sup
t
t
hξ|ηio
t>0
ξ∈Λbp
ξ∈Λbp
tξ →∞ ηt ∈Hξ

tξ →∞

=1−
Since

1
maxp∈P ωp (η)


S


=∞
η ∈ p∈P Liouvillep


S
S
max ωp (η) ∈ (1, ∞) η ∈ p∈P VWAp \ p∈P Liouvillep

p∈P

S

= 1
η∈
/ p∈P VWAp

applying (17.5.4) and (17.5.5) completes the proof.



We are now ready to state our main theorem regarding the relation between
exact dimensionality and dynamical Diophantine approximation:
Theorem 17.5.8. The following are equivalent:
(A) µ(VWAp ) = 0 ∀p ∈ P .

(B) µ is exact dimensional.
(C) dimH (µ) = δ.

(D) µ(VWAξ ) = 0 ∀ξ ∈ Λ.
The implication (B) ⇒ (C) is part of Proposition 17.5.3, while (C) ⇒ (D) is

an immediate consequence of Theorem 17.5.6, and (D) ⇒ (A) is trivial. Thus we
demonstrate (A) ⇒ (B):
S
Proof of (A) ⇒ (B). Fix η ∈ Λ \ p∈P VWAp and t > 0. Suppose that
ηt ∈ Hξ for some ξ ∈ Λbp . Let t− < t < t+ satisfy

/
t− ≍+ tξ , t+ ≍+ 2hξ|ηio − tξ , and ηt± ∈
Then by Lemma 17.3.7,
µ(B(η, e−t± )) ≍× e−δt± .

[
(H ).

308

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

In particular
(17.5.6)

1
.+ δt+ .
µ(B(η, e−t ))

δt− .+ log

Now, by Observation 17.5.7, we have
2(hξ|ηio − tξ + (constant))
t+ − t−
→ 0 as t → ∞.
≤
t
tξ
Since t− < t < t+ , it follows that t− /t, t+ /t → 1 as t → ∞. Combining with
(17.5.6) gives dµ (η) = δ (cf. (17.5.1)). But by assumption (A), this is true for
µ-a.e. η ∈ Λ. Thus µ is exact dimensional.



17.5.2. Examples and non-examples of exact dimensional measures.
Combining Theorems 17.5.8 and 17.5.5 gives a necessary and sufficient condition
for µ to be exact dimensional in terms of the convergence or divergence of a family
of series. We can ask how often this condition is satisfied. Our first result shows
that it is almost always satisfied:
Theorem 17.5.9. If for all p ∈ P , the series
(17.5.7)

X

h∈Gp

e−δkhk khk ≍×

X

h∈Gp

log khkp ≍×
khk−2δ
p

∞
X

e−2δk kNp (ek )

k=0

converges, then µ is exact dimensional.
Proof. Fix p ∈ P and κ > 0. We have
X
X
e−δkhk
Σdiv (p, κ) =
e−δkgk
g∈G

=

h∈Gp
khk≥κkgk/2

X

e−δkhk

X

e−δkhk

h∈Gp

≍×
≤
.×

e−δkhk

≍×

h∈Gp

e−δk #{g ∈ G : k − 1 ≤ kgk < k}

X

k≤2khk/κ+1

e

−δkhk

e

−δkhk

h∈Gp

X

X

k≤2khk/κ+1

h∈Gp

X

e−δkgk

g∈G
kgk≤2khk/κ

h∈Gp

X

X

X

e−δk NX,G (k)
1

(by Corollary 16.7.1)

k≤2khk/κ+1

khk.

So if (17.5.7) converges, so does Σdiv (p, κ), and thus by Theorems 17.5.5 and 17.5.8,
µ is exact dimensional.

Corollary 17.5.10. If for all p ∈ P , δp < δ, then µ is exact dimensional.

17.5. EXACT DIMENSIONALITY OF µ

309

Proof. In this case, the series (17.5.7) converges, as it is dominated by Σs (Gp )
for any s ∈ (δp , δ).



Remark 17.5.11. Combining with Proposition 10.3.10 shows that if µ is not
exact dimensional, then
X
X
e−δkhk < ∞ =
e−δkhk khk
h∈Gp

h∈Gp

for some p ∈ P . Equivalently,
∞
X

k=0

e−2δk Np (ek ) < ∞ =

∞
X

e−2δk kNp (ek ).

k=0

This creates a very “narrow window” for the orbital counting function Np .
Corollary 17.5.12. If µ is doubling, then µ is exact dimensional.
Proof. If µ is doubling, then
∞
X

e−2δk kNp (ek ) =

k=0

=
≍×

∞ X
∞
X

e−2δ(k+ℓ) Np (ek+ℓ )

k=1 ℓ=0
∞
X
e−2δk Iep (ek )
k=1
∞
X

k=1

e−2δk Np (ek ).

Remark 17.5.11 completes the proof.

(by Proposition 17.4.4)


Our next theorem shows that in certain circumstances, the converse holds in
Theorem 17.5.9. Specifically:
Theorem 17.5.13. Suppose that X is an R-tree and that G is the pure Schottky
product (cf. Definition 14.5.7) of a parabolic group H with a lineal group J. Let p be
the global fixed point of H, so that P = {p} is a complete set of inequivalent parabolic
points for G (Proposition 12.4.19). Suppose that the series (17.5.7) diverges. Then
µ is not exact dimensional; moreover, µ(Liouvillep ) = 1 and dimH (µ) = 0.

Example 17.5.14. To see that the hypotheses of this theorem are not vacuous,
fix δ > 0 and let
R2δ
,
f (R) =
log2 (R)
P∞ −2δk
kf (ek )
or more generally, let f be any increasing function such that
1 e
P∞ −2δk
k
diverges but 1 e
f (e ) converges. By Theorem 14.1.5, there exists an R-tree

X and a parabolic group H ≤ Isom(X) such that NEp ,H ≍× f . Then the series

310

17. QUASICONFORMAL MEASURES OF GEOMETRICALLY FINITE GROUPS

(17.5.7) diverges, but Σδ (H) < ∞. Thus, there exists a unique r > 0 such that
2

∞
X

e−δr =

n=1

1
·
Σδ (H) − 1

Let J = rZ, interpreted as a group acting by translations on the R-tree R, and
P∞
let G be the pure Schottky product of H and J. Then Σδ (J) − 1 = 2 n=1 e−δr ,
so (Σδ (H) − 1)(Σδ (J) − 1) = 1. Since the map s 7→ (Σs (H) − 1)(Σs (J) − 1) is
decreasing, it follows from Proposition 14.5.8 that ∆(G) = [0, δ]. In particular, G

is of divergence type, so Standing Assumption 17.5.1 is satisfied.
Remark 17.5.15. Applying a BIM embedding allows us to construct an example acting on H∞ .
Proof of Theorem 17.5.13. As in the proof of Proposition 14.5.8 we let
E = (H \ {id})(J \ {id}),
so that
G=

[

JE n H.

n≥0

Define a measure θ on E via the formula
X
θ=
e−δkgk δg .
g∈E

By Proposition 14.5.8, the fact that G is of divergence type (Standing Assumption 17.5.1), and the fact that Σδ (J), Σδ (H) < ∞ (Proposition 10.3.10), θ is a
probability measure. The Patterson–Sullivan measure of G is related to θ by the
formula
X
1
µ=
e−δkjk j∗ π∗ [θN ],
Σδ (J) − 1
j∈J

where π : E N → ΛG is the coding map.
Next, we use a theorem proven independently by H. Kesten and A. Raugi,2

which we rephrase here in the language of measure theory:
Theorem 17.5.16 ([111]; see also [149]). Let θ be a probability measure on a
set E, and let f : E → R be a function such that
Z
|f (x)| dθ(x) = ∞.
N
Then for θN -a.e. (xn )∞
1 ∈R ,

|f (xn+1 )|
= ∞.
lim sup Pn
n→∞ |
1 f (xi )|

2We are grateful to “cardinal” of http://mathoverflow.net and J. P. Conze, respectively, for

these references.

17.5. EXACT DIMENSIONALITY OF µ

311

Letting f (g) = kgk, the theorem applies to our measure θ, because our assumpR
tion that (17.5.7) diverges is equivalent to the assertion that f (x) dθ(x) = ∞.
N
N
Now fix j ∈ J, and let (gn )∞
1 ∈ E be a θ -typical point. Then the limit point
η = lim jg1 · · · gn (o)
n→∞

represents a typical point with respect to the Patterson–Sullivan measure µ. By
Theorem 17.5.16, we have
kgn+1 k
= ∞.
lim sup Pn
n→∞
1 kgi k

Write gi = hi ji for each i. Then kgi k = khi k + kji k. Since
law of large numbers implies that

kjn+1 k
limn→∞ P
n
1 kji k

< ∞, so

R

kjkdθ(hj) < ∞, the

khn+1 k
= ∞.
lim sup Pn
n→∞
1 kgi k

But khn+1 k represents the length of the excursion of the geodesic [o, η] into the
cusp corresponding to the parabolic point g1 · · · gn (p). Combining with Observa-

tion 17.5.7 shows that η ∈ Liouvillep . Since η was a µ-typical point, this shows

that µ(Liouvillep ) = 1. By Theorem 17.5.6, this implies that dimH (µ) = 0. By
Observation 17.5.4, µ is not exact dimensional.


APPENDIX A

Open problems
Problem 1 (Cf. Chapter 8, Section 13.4). Do there exist a hyperbolic metric
e
e
space X and a group G such that δ(G)
< δ(G) = ∞, but δ(G)
cannot be “computed

from” the modified Poincaré exponent of a locally compact group via Definition
8.2.1? This question is vague because a more precise version might be contradicted
e
by Example 13.4.9, in which a group G is constructed such that δ(G)
< δ(G) = ∞

but the closure of G (in the compact-open topology) is not locally compact. In this
e
e
but there is still a locally compact group
case, δ(G)
cannot be computed from δ(G),
“hidden” in the argument, namely the closure of G ↿ Hd = ι1 (Γ). Is there any
Poincaré irregular group whose construction is not somehow “based on” a locally
compact group?

Problem 2 (Cf. Theorem 1.2.3). If G is a Poincaré irregular parabolic group,
e
does the modified Poincaré exponent δ(G)
have a geometric significance? Theorem
1.2.3 does not apply directly since G is elementary. It is tempting to claim that
(A.1)

e
δ(G)
= inf{dimH (Λr (H)) : H ≥ G nonelementary}

(under some reasonable hypotheses about the isometry group of the space in question), but it seems that the right hand side is equal to infinity in most cases due
to Proposition 10.3.7(iii). Note that by contrast, (A.1) is usually true for Poincaré
regular groups; for example, it holds in the Standard Case [18].

Problem 3 (Cf. Chapter 11, Remark 11.2.12). Given a virtually nilpotent
group Γ which is not virtually abelian, determine whether there exists a homomorphism Φ : Γ → Isom(B) such that δ(Φ(Γ)) = α(Γ)/2, where both quantities are
defined in Section 11.2. Intuitively, this corresponds to the existence an equivariant embedding of Γ into B which approaches infinity “as fast as possible”. It is
known [59, Theorem 1.3] that such an embedding cannot be quasi-isometric, but
this by itself does not imply the non-existence of a homomorphism with the desired
property.
313

314

A. OPEN PROBLEMS

Problem 4 (Cf. Chapter 11, Remark 11.2.14). Does there exist a strongly
discrete parabolic subgroup of Isom(H∞ ) isomorphic to the Heisenberg group which
has infinite Poincaré exponent? 1
Problem 5 (Cf. Chapter 12, Section 12.2). Is there any form of discreteness for which there exists a cobounded subgroup of Isom(H) (for example, UOT
discreteness)? If so, what is the strongest such form of discreteness?
Problem 6 (Cf. Chapter 17). Can Theorem 17.5.13 be improved as follows?
Conjecture. Let X be a hyperbolic metric space and let G ≤ Isom(X) be a
geometrically finite group such that for some p ∈ Λbp , the series (17.5.7) diverges.

Then the δ-quasiconformal measure µ is not exact dimensional.

What if some of the hypotheses of this conjecture are strengthened, e.g. X is
strongly hyperbolic (e.g. X = H∞ ), or G is a Schottky product of a parabolic group
with a lineal group?

1It has been pointed out to us by X. Xie that this question is answered affirmatively by [59,
Proposition 3.10], letting the function f in that proposition be any function whose growth is
sublinear.

APPENDIX B

Index of defined terms
See also Conventions 1-9 on pages xix, 21, and 59.
• acts irreducibly: Definition 7.6.1, p.122

• acts properly discontinuously: Definition 5.2.11, p.92
• acts reducibly: Definition 7.6.1, p.122
• algebraic hyperbolic space: Definition 2.2.5, p.7
• attracting fixed point : Definition 6.1.1, p.95
• ball model : §2.5.1, p.18
• bi-infinite geodesic: Definition 4.4.2, p.67

• BIM embedding: Definition 13.1.3, p.224
• BIM representation: Definition 13.1.3, p.224

• bordification: Definition 3.4.2, p.34
• ξ-bounded : Definition 12.3.1, p.201

• bounded parabolic point : Definition 12.3.4, p.202

• Busemann function: (3.3.5), p.31
• CAT(-1) inequality: (3.2.1), p.28

• CAT(-1) space: Definition 3.2.1, p.28
• Cayley graph: Example 3.1.2, p.24

• Cayley hyperbolic plane: Remark 2.1.1, p.3
• Cayley metric: Example 3.1.2, p.24

• center (of a triangle in an R-tree): Definition 3.1.11, p.26
• center (of a horoball): Definition 12.1.1, p.193

• cobounded : Definition 12.2.1, p.196
• codoubling (function): Definition 17.4.2, p.295

• convergence type: Definition 8.1.4, p.130
• compact-open topology (COT): p.83

• compact type, semigroup of : Definition 7.7.1, p.123
• comparison point : p.27, Definition 4.4.12, p.72

• comparison triangle: Example 3.1.9, p.25; Definition 4.4.12, p.72

• compatible (regarding a metametric and a topology): Definition 3.6.4, p.49
• complete set of inequivalent parabolic points: Definition 12.4.13, p.213
• cone: (14.1.1), p.233

315

316

B. INDEX OF DEFINED TERMS

• conformal measure: Definition 15.1.1, p.255

• conical convergence: p.109
• connected graph: Definition 3.1.1, p.23

• contractible cycles (property of a graph): Definition 14.2.1, p.237
• convex-cobounded : Definition 12.2.5, p.198
• convex hull : Definition 7.5.1, p.119
• convex : (7.5.1), p.119

• convex core: Definition 7.5.7, p.121
• cycle: (3.1.4), p.25

• Dirichlet domain: Definition 12.1.4, p.194
• divergence type: Definition 8.1.4, p.130

• domain of reflexivity: Definition 3.6.1, p.48
• doubling (metric space): Footnote 2, p.187

• doubling (function): Definition 17.4.2, p.295

• doubling (measure): Section 17.4, p.295
• dynamical derivative: Proposition 4.2.12, p.63

• Edelstein-type isometry: Definition 11.1.11, p.180
• elementary: Definition 7.3.2, p.114
• elliptic isometry: Definition 6.1.2, p.95
• elliptic semigroup: Definition 6.2.2, p.99

• ergodic: Definition 15.3.1, p.257
• equivalent (for Gromov sequences): Definition 3.4.1, p.34

• extended visual metric: Proposition 3.6.13, p.52
• fixed point (neutral/attracting/repelling): Definition 6.1.2, p.95
• fixed point (parabolic): Definition 6.2.7, p.100
• focal semigroup: Definition 6.2.13, p.101
• free group: Remark 10.1.1, p.158
• free product : Section 10.1, p.157

• free semigroup: Remark 10.1.1, p.158

• general type, semigroup of : Definition 6.2.13, p.101
• generalized convergence type: Definition 8.2.3, p.132

• generalized divergence type: Definition 8.2.3, p.132
• generalized polar coordinate functions: Definition 4.6.1, p.79
• geodesic metric space: Remark 3.1.5, p.24
• geodesic segment : Remark 3.1.5, p.24

• geodesic triangle: p.27, Definition 4.4.12, p.72
• geometric product : Example 14.5.10, p.249
• geodesic path: Section 14.2, p.237
• geodesic ray/line: Definition 4.4.2, p.67

B. INDEX OF DEFINED TERMS

• geometric realization: Definition 3.1.1, p.23

• geometric graph: Definition 3.1.1, p.23
• geometrically finite: Definition 12.4.1, p.206
• Gromov boundary: Definition 3.4.2, p.34
• Gromov hyperbolic: Definition 3.3.2, p.30
• Gromov’s inequality: (3.3.4), p.30
• Gromov product : (3.3.2), p.30

• Gromov sequence: Definition 3.4.1, p.34
• Gromov triple: Definition 4.1.1, p.59

• global fixed points: Notation 6.2.1, p.99
• growth rate: (11.2.2), p.184; Definition 17.4.6, p.299

• global Schottky product : Definition 10.2.1, p.158
• group of isometries: p.8

• Haagerup property: §11.1.1, p.176
• half-space: Remark 10.2.5, p.160
• half-space model : §2.5.2, p.19

• horoball : Definition 12.1.1, p.193
• horospherical convergence: Definition 7.1.3, p.111
• horospherical limit set : Definition 7.2.1, p.112
• hyperbolic: Definition 3.3.2, p.30
• hyperboloid model : §2.2, p.4
• implied constant : Convention 1, p.xix

• inward focal : Definition 6.2.15, p.102
• irreducible action: Definition 7.6.1, p.122

• isomorphism (between pairs (X, bord X) and (Y, bord Y )): p.14
• length spectrum: Remark 13.1.6, p.225

• limit set (of a semigroup): Definition 7.2.1, p.112
• limit set (of a partition structure): Definition 9.1.7, p.137
• lineal semigroup: Definition 6.2.13, p.101
• Lorentz boosts: (2.3.3), p.8
• lower central series: §11.2.1, p.183

• lower polynomial growth rate: Definition 17.4.6, p.299
• loxodromic isometry: Definition 6.1.2, p.95

• loxodromic semigroup: Definition 6.2.2, p.99
• Margulis’s lemma: Proposition 11.1.3, p.176
• metametric: Definition 3.6.1, p.48
• metric derivative: p.60, p.61

• moderately discrete (MD): Definition 5.2.1, p.87
• modified Poincaré exponent : Definition 8.2.3, p.132

317

318

B. INDEX OF DEFINED TERMS

• natural action: (on a Cayley graph) Remark 3.1.3, p.24
• natural map (from a free product): Section 10.1, p.157
• ρ-net : Footnote 3, p.133

• neutral fixed point : Definition 6.1.1, p.95
• nilpotent : §11.2.1, p.183

• nilpotency class: §11.2.1, p.183
• nonelementary: Definition 7.3.2, p.114

• orbital counting function: Remark 8.1.3, p.129
• outward focal : Definition 6.2.15, p.102
• parabolic isometry: Definition 6.1.2, p.95
• parabolic fixed point : Definition 6.2.7, p.100

• parabolic semigroup: Definition 6.2.2, p.99
• parameterization (of a geodesic): Remark 3.1.5, p.24
• partition structure: Definition 9.1.4, p.137

• path: Section 14.2, p.237
• path metric: Definition 3.1.1, p.23, (14.4.1), p.241

• Poincaré exponent : Definition 8.1.1, p.129
• Poincaré extension: Observation 2.5.6, p.20
• Poincaré integral : (8.2.1), p.131
• Poincaré regular/irregular : p.134

• Poincaré set : Notation 8.1.7, p.130
• Poincaré series: Definition 8.1.1, p.129

• polynomial growth rate: (11.2.2), p.184; Definition 17.4.6, p.299
• pre-doubling (parabolic group): Definition 17.4.9, p.301
• proper : Remark 1.1.3, p.xxii
• properly discontinuous (PrD): Definition 5.2.11, p.92
• pure Schottky product : Definition 14.5.7, p.247
• quasiconformal measure: Definition 15.1.1, p.255
• quasiconvex core: Definition 7.5.7, p.121

• quasi-isometry/quasi-isometric: Definition 3.3.9, p.33
• radial convergence: Definition 7.1.2, p.110
• radial limit set : Definition 7.2.1, p.112
• Radon: Remark 16.3.2, p.271

• rank (of an abelian group): §11.2.1, p.183
• reducible action: Definition 7.6.1, p.122

• regularly geodesic: Definition 4.4.5, p.67
• repelling fixed point : Definition 6.1.1, p.95

• Samuel–Smirnov compactification: Proposition 16.1.1, p.267
• Schottky group: Definition 10.2.4, p.159

B. INDEX OF DEFINED TERMS

319

• Schottky position: Definition 10.2.1, p.158

• Schottky product : Definition 10.2.1, p.158
• Schottky semigroup: Definition 10.2.4, p.159

• Schottky system: Definition 10.2.1, p.158
• ρ-separated set : Footnote 1, p.131

• sesquilinear form: p.3
• shadow : Definition 4.5.1, p.74

• similarity: Observation 2.5.6, p.20
• simplicial tree: Definition 3.1.7, p.25

• F-skew linear : (2.3.4), p.9
• skew-symmetric: p.3

• Standard Case: Convention 9, p.60
• standard parameterization: p.70

• stapled union: Definition 14.4.1, p.240

• strong operator topology (SOT): p.83
• strongly discrete (SD): Definition 5.2.1, p.87, Remark 8.2.5, p.133

• strongly (Gromov) hyperbolic: Definition 3.3.6, p.32
• strongly separated Schottky group/product/system: Definition 10.3.1, p.160
• substructure (of a partition structure): Definition 9.1.5, p.137
• s-thick : Definition 9.1.4, p.137
• topological discreteness: Definition 5.2.6, p.89
• totally geodesic subset : Definition 2.4.2, p.14
• tree, simplicial : Definition 3.1.7, p.25
• tree (on N): Definition 9.1.2, p.136

• R-tree: Definition 3.1.10, p.25
• Z-tree: Definition 3.1.7, p.25

• tree-geometric: Definition 14.5.2, p.245
• tree triangle: p.27

• Tychonoff topology: p.84

• uniform operator topology (UOT): p.83
• uniformly radial convergence: Definition 7.1.2, p.110

• uniformly radial limit set : Definition 7.2.1, p.112
• uniquely geodesic metric space: Remark 3.1.5, p.24

• unweighted simplicial tree: Definition 3.1.7, p.25
• upper polynomial growth rate: Definition 17.4.6, p.299
• virtually nilpotent : §11.2.1, p.183
• visual metric: p.50

• weakly discrete (WD): Definition 5.2.1, p.87
• weakly separated Schottky group/product/system: Definition 10.3.1, p.160

320

B. INDEX OF DEFINED TERMS

• weighted Cayley graph: Example 3.1.2, p.24

• weighted undirected graph: Definition 3.1.1, p.23

Bibliography
1. I. Adeboye and G. Wei, On volumes of hyperbolic orbifolds, Algebr. Geom. Topol. 12 (2012),
no. 1, 215–233.
2. Clemens Adelmann and Eberhard H.-A. Gerbracht, Letters from William Burnside to Robert
Fricke: automorphic functions, and the emergence of the Burnside problem, Arch. Hist.
Exact Sci. 63 (2009), no. 1, 33–50. MR 2470902
3. I. Agol, Tameness of hyperbolic 3-manifolds, http://arxiv.org/abs/math/0405568, preprint
2004.
4. L. V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures
in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981.
5. T. Akaza, Poincaré theta series and singular sets of Schottky groups, Nagoya Math. J. 24
(1964), 43–65.
6. A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications,
Trudy Mat. Inst. Steklov., v 38, Trudy Mat. Inst. Steklov., v 38, Izdat. Akad. Nauk SSSR,
Moscow, 1951, pp. 5–23.
7. D. Allcock, Reflection groups on the octave hyperbolic plane, J. Algebra 213 (1999), no. 2,
467–498.
8. Roger C. Alperin, An elementary account of Selberg’s lemma, Enseign. Math. (2) 33 (1987),
no. 3-4, 269–273. MR 925989
9. A. Ancona, Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature, Rev. Mat. Iberoamericana 10 (1994), no. 1, 189–220.
10. J. W. Anderson, K. Falk, and P. Tukia, Conformal measures associated to ends of hyperbolic
n-manifolds, Q. J. Math. 58 (2007), no. 1, 1–15.
11. M. T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J.
Differential Geom. 18 (1983), no. 4, 701–721.
12. B. N. Apanasov, Discrete groups in space and uniformization problems. Translated and
revised from the 1983 Russian original, Mathematics and its Applications (Soviet Series),
40, Kluwer Academic Publishers Group, Dordrecht, 1991.
13. C. J. Atkin, The Hopf-Rinow theorem is false in infinite dimensions, Bull. Lond. Math. Soc.
7 (1975), no. 3, 261–266.
14. J. C. Baez, The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145–205.
15. W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature, Progress
in Mathematics, 61, Birkhäuser Boston, Inc., Boston, MA, 1985.
16. H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London
Math. Soc. (3) 25 (1972), 603–614.
17. A. F. Beardon, The Hausdorff dimension of singular sets of properly discontinuous groups,
Amer. J. Math. 88 (1966), 722–736.
18.

, The exponent of convergence of Poincaré series, Proc. London Math. Soc. (3) 18
(1968), 461–483.
321

322

19.
20.

BIBLIOGRAPHY

, Inequalities for certain Fuchsian groups, Acta Math. (1971), 221–258.
, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, SpringerVerlag, New York, 1983.

21. A. F. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental
polyhedra, Acta Math. 132 (1974), 1–12.
22. M. E. B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s property (T), New Mathematical
Monographs, 11, Cambridge University Press, Cambridge, 2008.
23. M. V. Belolipetsky, Hyberbolic orbifolds of small volume, http://arxiv.org/abs/1402.5394,
preprint 2014.
24. V. I. Bernik and M. M. Dodson, Metric Diophantine approximation on manifolds, Cambridge
Tracts in Mathematics, vol. 137, Cambridge University Press, Cambridge, 1999.
25. L. Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2)
91 (1970), 570–600.
26. M. Bestvina, The topology of Out(Fn ), Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 373–384.
27. M. Bestvina and M. Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256
(2014), 104–155.
28. C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179
(1997), no. 1, 1–39.
29. S. Blachère, P. Haı̈ssinsky, and P. Mathieu, Harmonic measures versus quasiconformal measures for hyperbolic groups, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 4, 683–721.
30. J. Blanc and S. Cantat, Dynamical degrees of birational transformations of projective surfaces, http://arxiv.org/abs/1307.0361, preprint 2013.
31. M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal.
10 (2000), no. 2, 266–306.
32. A. Borbély, The nonsolvability of the Dirichlet problem on negatively curved manifolds,
Differential Geom. Appl. 8 (1998), no. 3, 217–237.
33. M. Bourdon, Structure conforme au bord et flot géodésique d’un CAT(-1)-espace (Conformal
structure at the boundary and geodesic flow of a CAT(-1)-space), Enseign. Math. (2) 41
(1995), no. 1–2, 63–102 (French).
34. B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993),
no. 2, 245–317.
35.

, Some results on the geometry of convex hulls in manifolds of pinched negative
curvature, Comment. Math. Helv. 69 (1994), no. 1, 49–81.

36.

, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995),
no. 1, 229–274.

37.

, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012), no. 3, 66 pp.

38. E. Breuillard, B. Green, and T. C. Tao, Approximate subgroups of linear groups, Geom.
Funct. Anal. 21 (2011), no. 4, 774–819.
39. M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der
Mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999.
40. M. Burger, A. Iozzi, and N. Monod, Equivariant embeddings of trees into hyperbolic spaces,
Int. Math. Res. Not. (2005), no. 22, 1331–1369.
41. M. Burger and S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators, J.
Amer. Math. Soc. 9 (1996), no. 1, 57–93.

BIBLIOGRAPHY

323

42. S. Buyalo, Geodesics in Hadamard spaces, Algebra i Analiz 10 (1998), no. 2, 93–123 (Russian), translation in St. Petersburg Math. J. 10 (1999), no. 2, 293–313.
43. D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifold, J. Amer.
Math. Soc. 19 (2006), no. 2, 385–446.
44. J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom.
Dedicata 16 (1984), no. 2, 123–148.
45. J. W. Cannon, W. J. Floyd, R. W. Kenyon, and W. R. Parry, Hyperbolic geometry, Flavors
of geometry, Math. Sci. Res. Inst. Publ., vol. 31, Cambridge Univ. Press, Cambridge, 1997,
pp. 59–115.
46. S. Cantat, The Cremona group in two variables, Proceedings of the sixth European Congress of Math., pp. 211-225 (Europ. Math. Soc., 2013), 2013, available at
http://perso.univ-rennes1.fr/serge.cantat/Articles/ecm.pdf.
47. S. Cantat and S. Lamy, Normal subgroups in the Cremona group, with an appendix by Yves
de Cornulier, Acta Math. 210 (2013), no. 1, 31–94.
48. P.-E. Caprace, Y. de Cornulier, N. Monod, and R. Tessera, Amenable hyperbolic groups,
http://arxiv.org/abs/1202.3585v1, preprint 2012.
49. C. Champetier, Propriétés statistiques des groupes de présentation finie (statistical properties of finitely presented groups), Adv. Math. 116 (1995), no. 2, 197–262 (French).
50. P.-A. Cherix, M. G. Cowling, P. Jolissaint, P. Julg, and A. Valette, Groups with the Haagerup
property: Gromov’s a-T-menability, Progress in Mathematics, vol. 197, Birkhäuser Verlag,
Basel, 2001.
51. I. M. Chiswell, Introduction to Λ-trees, World Scientific Publishing Co., Inc., River Edge,
NJ, 2001.
52. F. Choucroun, Arbres, espaces ultramétriques et bases de structure uniforme (Trees, ultrametric spaces and bases with uniform structure), Geom. Dedicata 53 (1994), no. 1, 69–74
(French).
53. M. Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens
de Gromov (Patterson-Sullivan measures on the boundary of a hyperbolic space in the sense
of Gromov), Pacific J. Math. 159 (1993), no. 2, 241–270 (French).
54. K. Corlette and A. Iozzi, Limit sets of discrete groups of isometries of exotic hyperbolic
spaces, Trans. Amer. Math. Soc. 351 (1999), no. 4, 1507–1530.
55. F. Dal’bo, J.-P. Otal, and M. Peigné, Séries de Poincaré des groupes géométriquement
finis. (Poincaré series of geometrically finite groups), Israel J. Math. 118 (2000), 109–124
(French).
56. F. Dal’bo, M. Peigné, J.-C. Picaud, and A. Sambusetti, On the growth of quotients of
Kleinian groups, Ergodic Theory Dynam. Systems 31 (2011), no. 3, 835–851.
57. Tushar Das, David Simmons, and Mariusz Urbański, Geometry and dynamics in Gromov
hyperbolic metric spaces II: Thermodynamic formalism, in preparation.
58. Tushar Das, Bernd O. Stratmann, and Mariusz Urbański, The Bishop–Jones relation and the
Hausdorff geometry of convex-cobounded limit sets in infinite-dimensional hyperbolic space,
http://www.urbanskimath.com/wp-content/uploads/2014/07/DaStUrBishopJones_2014_07_23C.pdf,
preprint 2014, to appear in Stoch. Dyn.
59. Y. de Cornulier, R. Tessera, and A. Valette, Isometric group actions on Hilbert spaces:
growth of cocycles, Geom. Funct. Anal. 17 (2007), no. 3, 770–792.
60. P. de la Harpe, Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert
space, Lecture Notes in Mathematics, Vol. 285, Springer-Verlag, Berlin-New York, 1972.

324

BIBLIOGRAPHY

, Topics in geometric group theory, Chicago Lectures in Mathematics, University of

61.

Chicago Press, Chicago, IL, 2000.
62. P. de la Harpe and A. Valette, La propriété (T) de Kazhdan pour les groupes localement
compacts (avec un appendice de Marc Burger) (Kazhdan’s property (T) for locally compact
groups (with an appendix by Marc Burger)), Astérisque 175 (1989), 158 (French).
63. H. P. de Saint-Gervais, Uniformisation des surfaces de Riemann. Retour sur un théorème
centenaire (Uniformization of Riemann surfaces. A look back at a 100-year-old theorem),
ENS Éditions, Lyon, 2010 (French), The name of Henri Paul de Saint-Gervais covers a
group composed of fifteen mathematicians : A. Alvarez, C. Bavard, F. Béguin, N. Bergeron,
M. Bourrigan, B. Deroin, S. Dumitrescu, C. Frances, É. Ghys, A. Guilloux, F. Loray, P.
Popescu-Pampu, P. Py, B. Sévennec, and J.-C. Sikorav.
64. M. Dehn, Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911), no. 1, 116–144
(German).
65. T. Delzant and P. Py, Kähler groups, real hyperbolic spaces and the Cremona group, Compos.
Math. 148 (2012), no. 1, 153–184.
66. G. Dimitrov, F. Haide, L. Katzarkov, and M. Kontsevich, Dynamical systems and categories,
http://arxiv.org/abs/1307.8418, preprint 2013.
67. B. Duchesne, Infinite-dimensional nonpositively curved symmetric spaces of finite rank, Int.
Math. Res. Not. IMRN (2013), no. 7, 1578–1627.
, Infinite dimensional riemannian symmetric spaces with fixed-sign curvature opera-

68.

tor, http://arxiv.org/abs/1109.0441, preprint 2012.
69. P. B. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109.
70. M. Edelstein, On non-expansive mappings of Banach spaces, Math. Proc. Cambridge Philos.
Soc. 60 (1964), 439–447.
71. K. J. Falconer and D. T. Marsh, On the Lipschitz equivalence of Cantor sets, Mathematika
39 (1992), no. 2, 223–233.
72. J. L. Fernández and M. V. Melián, Bounded geodesics of Riemann surfaces and hyperbolic
manifolds, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3533–3549.
73. Lior

Fishman,

imation

and

David
the

Simmons,

geometry

of

and

limit

Mariusz

sets

in

Urbański,

Gromov

Diophantine

hyperbolic

metric

approxspaces,

http://arxiv.org/abs/1301.5630, preprint 2013, to appear Mem. Amer. Math. Soc.
74. G. B. Folland, Real analysis: Modern techniques and their applications, Pure and Applied
Mathematics (New York), John Wiley & Sons, Inc., New York, 1984.
75. V. P. Fonf and J. Lindenstrauss, Some results on infinite-dimensional convexity, Israel J.
Math. 108 (1998), 13–32.
76. M. I. Garrido Carballo, J. Á. Jaramillo, and Y. C. Rangel, Algebras of differentiable functions
on Riemannian manifolds, Bull. Lond. Math. Soc. 41 (2009), no. 6, 993–1001.
77. É Ghys and P. de la Harpe, Espaces métriques hyperboliques (hyperbolic metric spaces),
Sur les groupes hyperboliques d’aprés Mikhael Gromov (Bern, 1988), Progr. Math., 83,
Birkhäuser Boston, Boston, MA, 1990, pp. 27–45 (French).
78. W. M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs. Oxford
Science Publications, The Clarendon Press, Oxford University Press, New York, 1999.
79. J. J. Gray, Linear differential equations and group theory from Riemann to Poincaré, Modern
Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008.
, Henri Poincaré. A scientific biography, Princeton University Press, Princeton, NJ,

80.
2013.

BIBLIOGRAPHY

325

81. L. Greenberg, Discrete subgroups of the Lorentz group, Math. Scand. 10 (1962), 85–107.
82. G. Greschonig and K. Schmidt, Ergodic decomposition of quasi-invariant probability measures, Colloq. Math. 84/85 (2000), part 2, 493–514.
83. M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics:
Proceedings of the 1978 Stony Brook conference (State Univ. New York, Stony Brook, N.Y.,
1978), Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–
213.
, Infinite groups as geometric objects, Proceedings of the International Congress of

84.

Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 385–392.
, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8,

85.

Springer, New York, 1987, pp. 75–263.
86.

, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex,
1991), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge,
1993, pp. 1–295.

87. Y. Guivarc’h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B 271
(1970), A237–A239 (French).
88. M. Hamann,

Group actions on metric spaces:

fixed points and free subgroups,

http://arxiv.org/abs/1301.6513.
89. M. Handel and L. Mosher, The free splitting complex of a free group, I: hyperbolicity, Geom.
Topol. 17 (2013), no. 3, 1581–1672.
90. G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J. 2 (1936), no. 3,
530–542.
91. J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York,
2001.
92.

, What is ... a quasiconformal mapping?, Notices Amer. Math. Soc. 53 (2006), no.
11, 1334–1335.

93. E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23–34.
94. S. Helgason, Differential geometry, Lie groups, and symmetric spaces., Pure and Applied
Mathematics, 80, Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), New YorkLondon, 1978.
95. S. W. Hensel, P. Przytycki, and R. C. H. Webb, Slim unicorns and uniform hyperbolicity for
arc graphs and curve graphs, http://arxiv.org/abs/1301.5577, preprint 2013.
96. A. Hilion and C. Horbez, The hyperbolicity of the sphere complex via surgery paths,
http://arxiv.org/abs/1210.6183, preprint 2012.
97. S. Hong, Patterson-Sullivan measure and groups of divergence type, Bull. Korean Math. Soc.
30 (1993), no. 2, 223–228.
98.

, Conical limit points and groups of divergence type, Trans. Amer. Math. Soc. 346
(1994), no. 1, 341–357.

99. E. Hopf, Fuchsian groups and ergodic theory, Trans. Amer. Math. Soc. 39 (1936), no. 2,
299–314.
100.

, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber.
Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304 (German).

101. P. Jolissaint, Borel cocycles, approximation properties and relative property T., Ergodic
Theory Dynam. Systems 20 (2000), no. 2, 483–499.
102. T. Jørgensen, Compact 3-manifolds of constant negative curvature fibering over the circle,
Ann. of Math. (2) 106 (1977), no. 1, 61–72.

326

BIBLIOGRAPHY

103. V. A. Kaimanovich, Bowen-Margulis and Patterson measures on negatively curved compact
manifolds, Dynamical systems and related topics (Nagoya, 1990), Adv. Ser. Dynam. Systems,
9, World Sci. Publ., River Edge, NJ, 1991, pp. 223–232.
104. I. Kapovich and K. Rafi, On hyperbolicity of free splitting and free factor complexes, Groups
Geom. Dyn. 8 (2014), no. 2, 391–414.
105. M. Kapovich, On the absence of Sullivan’s cusp finiteness theorem in higher dimensions,
Algebra and analysis (Irkutsk, 1989), 77-89, Amer. Math. Soc. Transl. Ser. 2, 163, Amer.
Math. Soc., Providence, RI, 1995.
106.

, Kleinian groups in higher dimensions, Geometry and dynamics of groups and spaces,
Progr. Math., vol. 265, Birkhäuser, Basel, 2008, pp. 487–564.

107. A. Karlsson, Nonexpanding maps and Busemann functions, Ergodic Theory Dynam. Systems
21 (2001), 1447–1457.
108. S. R. Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press,
Chicago, IL, 1992.
109. A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156,
Springer-Verlag, New York, 1995.
110. R. Kellerhalls, Volumes of cusped hyperbolic manifolds, Topology 37 (1998), no. 4, 719–734.
111. H. Kesten, Problems and solutions: Solutions of advanced problems: 5716, Amer. Math.
Monthly 78 (1971), no. 3, 305–308.
112. I. Kim, Length spectrum in rank one symmetric space is not arithmetic, Proc. Amer. Math.
Soc. 134 (2006), no. 12, 3691–3696.
113. V. L. Klee, Dispersed Chebyshev sets and coverings by balls, Math. Ann. 257 (1981), no. 2,
251–260.
114.

, Do infinite-dimensional Banach spaces admit nice tilings?, Studia Sci. Math. Hungar. 21 (1986), no. 3–4, 415–427.

115. F. Klein, Neue beiträge zur riemann’schen funktionentheorie, Math. Ann. 21 (1883), 141–214
(German).
116. D. Y. Kleinbock and B. Weiss, Badly approximable vectors on fractals, Israel J. Math. 149
(2005), 137–170.
117. P. Koebe, Riemannsche Mannigfaltigkeiten und nichteuklidische Raumformen, Sitzungsberichte Akad. Berlin 1930 (1930), 304–364, 505–541 (German).
118. J. Kurzweil, A contribution to the metric theory of diophantine approximations, Czechoslovak Math. J. 1(76) (1951), 149–178.
119. S. Lang, Fundamentals of differential geometry, Graduate Texts in Mathematics, vol. 191,
Springer-Verlag, New York, 1999.
120. G. Levitt, Graphs of actions on R-trees, Comment. Math. Helv. 69 (1994), no. 1, 28–38.
121. L. Li, K. Ohshika, and X. T. Wang, On Klein-Maskit combination theorem in space I, Osaka
J. Math. 46 (2009), no. 4, 1097–1141.
122. T. Lundh, Geodesics on quotient manifolds and their corresponding limit points, Michigan
Math. J. 51 (2003), no. 2, 279–304.
123. J. M. Mackay and J. T. Tyson, Conformal dimension: Theory and application, University
Lecture Series, 54, American Mathematical Society, Providence, RI, 2010.
124. J. H. Maddocks, Restricted quadratic forms and their application to bifurcation and stability
in constrained variational principles, SIAM J. Math. Anal. 16 (1985), no. 1, 47–68.

BIBLIOGRAPHY

327

125. Y. I. Manin, Cubic forms: algebra, geometry, arithmetic. Translated from the Russian by
M. Hazewinkel. North-Holland Mathematical library, vol. 4, North-Holland Publishing Co.,
Amsterdam-London; American Elsevier Publishing Co., New York, 1974.
126. H. A. Masur and S. Schleimer, The geometry of the disk complex, J. Amer. Math. Soc. 26
(2013), no. 1, 1–62.
127. R. D. Mauldin, T. Szarek, and Mariusz Urbański, Graph directed Markov systems on Hilbert
spaces, Math. Proc. Cambridge Philos. Soc. 147 (2009), 455–488.
128. R. D. Mauldin and Mariusz Urbański, Dimensions and measures in infinite iterated function
systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105–154.
, Graph directed Markov systems: Geometry and dynamics of limit sets, Cambridge

129.

Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003.
130. A. D. Michal, General differential geometries and related topics, Bull. Amer. Math. Soc. 45
(1939), 529–563.
, Infinite dimensional differential metrics with constant curvature, Proc. Nat. Acad.

131.

Sci. U. S. A. 34 (1948), 17–21.
132. N. Monod and P. Py, An exotic deformation of the hyperbolic space, Amer. J. Math. 136
(2014), no.5, 1249–1299.
133. G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies,
No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973.
134. S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann.
of Math. (2) 40 (1939), no. 2, 400–416.
135. P. J. Myrberg, Die Kapazität der singulären Menge der linearen Gruppen, Ann. Acad. Sci.
Fennicae. Ser. A. I. Math.-Phys. 1941 (1941), no. 10, 19 pp. (German).
136. S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge Tracts in Mathematics
and Mathematical Physics, No. 59, Cambridge University Press, London-New York, 1970.
137. F. Newberger, On the Patterson-Sullivan measure for geometrically finite groups acting on
complex or quaternionic hyperbolic space (Special volume dedicated to the memory of Hanna
Miriam Sandler (1960-1999)), Geom. Dedicata 97 (2003), 215–249.
138. P. J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture
Note Series, vol. 143, Cambridge University Press, Cambridge, 1989.
139. A. Y. Ol’shanskiı̆, Almost every group is hyperbolic, Internat. J. Algebra Comput. 2 (1992),
no. 1, 1–17.
140. D. V. Osin, Acylindrically hyperbolic groups, http://arxiv.org/abs/1304.1246, preprint
2013.
141. P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de
rang un (Carnot-Carathéodory metrics and quasi-isometries of rank one symmetric spaces),
Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French).
142. S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3–4, 241–273.
143.

, Further remarks on the exponent of convergence of Poincaré series, Tohoku Math.
J. (2) 35 (1983), no. 3, 357–373.

144. F. Paulin, On the critical exponent of a discrete group of hyperbolic isometries, Differential
Geom. Appl. 7 (1997), no. 3, 231–236.
145. S. D. Pauls, The large scale geometry of nilpotent Lie groups, Comm. Anal. Geom. 9 (2001),
no. 5, 951–982.
146. D. Preiss, Tilings of Hilbert spaces, Mathematika 56 (2010), no. 2, 217–230.

328

BIBLIOGRAPHY

147. J.-F.

Quint,

An

overview

of

Patterson–Sullivan

theory,

www.math.u-bordeaux1.fr/~ jquint/publications/courszurich.pdf.
148. J. G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol.
149, Springer, New York, 2006.
149. A. Raugi, Dépassement des sommes partielles de v.a.r. indépendantes équidistribuées sans
moment d’ordre 1 (Exceedance of partial sums of independent uniformly distributed real
random variables with undefined mean), Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 4,
723–734 (French).
150. R. H. Riedi and B. B. Mandelbrot, Multifractal formalism for infinite multinomial measures,
Adv. in Appl. Math. 16 (1995), no. 2, 132–150.
151. T. Roblin, Ergodicité et équidistribution en courbure négative (Ergodicity and uniform distribution in negative curvature), Mém. Soc. Math. Fr. (N. S.) 95 (2003), vi+96 pp. (French).
152.

, Un théoréme de Fatou pour les densités conformes avec applications aux revetements galoisiens en courbure négative (A Fatou theorem for conformal densities with applications to Galois coverings in negative curvature), Israel J. Math. 147 (2005), 333–357
(French).

153. B. Schapira, Lemme de l’ombre et non divergence des horosphéres d’une variété
géométriquement finie. (The shadow lemma and nondivergence of the horospheres of a geometrically finite manifold), Ann. Inst. Fourier (Grenoble) 54 (2004), no. 4, 939–987 (French).
154. V. Schroeder, Quasi-metric and metric spaces, Conform. Geom. Dyn. 10 (2006), 355–360.
155. Y. Shalom and T. C. Tao, A finitary version of Gromov’s polynomial growth theorem, Geom.
Funct. Anal. 20 (2010), no. 6, 1502–1547.
156. T. A. Springer and F. D. Veldkamp, Elliptic and hyperbolic octave planes. I, II, III., Nederl.
Akad. Wetensch. Proc. Ser. A 66=Indag. Math. 25 (1963), 413–451.
157. A. N. Starkov, Fuchsian groups from the dynamical viewpoint, J. Dynam. Control Systems
1 (1995), no. 3, 427–445.
158. B. O. Stratmann, The Hausdorff dimension of bounded geodesics on geometrically finite
manifolds, Ergodic Theory Dynam. Systems 17 (1997), no. 1, 227–246.
159. B. O. Stratmann and Mariusz Urbański, Pseudo-Markov systems and infinitely generated
Schottky groups, Amer. J. Math. 129 (2007), 1019–1062.
160. B. O. Stratmann and S. L. Velani, The Patterson measure for geometrically finite groups
with parabolic elements, new and old, Proc. London Math. Soc. (3) 71 (1995), no. 1, 197–220.
161. D. P. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes
Études Sci. Publ. Math. 50 (1979), 171–202.
162.
163.

, A finiteness theorem for cusps, Acta Math. 147 (1981), no. 3–4, 289–299.
, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions,
Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook conference (State
Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., vol. 97, Princeton Univ.
Press, Princeton, N.J., 1981, pp. 465–496.

164.

, Seminar on conformal and hyperbolic geometry, 1982, Notes by M. Baker and J.
Seade, IHES.

165.

, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite
Kleinian groups, Acta Math. 153 (1984), no. 3–4, 259–277.

166. S. J. Taylor, On the connexion between Hausdorff measures and generalized capacity, Proc.
Cambridge Philos. Soc. 57 (1961), 524–531.

BIBLIOGRAPHY

329

167. J. Tits, A “theorem of Lie–Kolchin” for trees, Contributions to algebra (collection of papers
dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 377–388.
168. D. A. Trotsenko and J. Väisälä, Upper sets and quasisymmetric maps, Ann. Acad. Sci. Fenn.
Math. 24 (1999), no. 2, 465–488.
169. M. Tsuji, Theory of conformal mapping of a multiply connected domain. III, Jap. J. Math.
19 (1944), 155–188.
170. P. Tukia, On isomorphisms of geometrically finite Möbius groups, Inst. Hautes Études Sci.
Publ. Math. 61 (1985), 171–214.
, The Poincaré series and the conformal measure of conical and Myrberg limit points,

171.

J. Anal. Math. 62 (1994), 241–259.
172. J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), no. 3, 187–231.
173.

, Hyperbolic and uniform domains in Banach spaces, Ann. Acad. Sci. Fenn. Math.
30 (2005), no. 2, 261–302.

174. A.

Valette,

Affine

isometric

actions

on

Hilbert

spaces

and

amenability,

http://www2.unine.ch/files/content/sites/math/files/shared/documents/articles/ESIValette.pdf,
preprint.
175. C. B. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative
curvature, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4965–5005.
176. Z. Yûjôbô, A theorem on Fuchsian groups, Math. Japonicae 1 (1949), 168–169.

