HYPERBOLIC ISOMETRIES AND BOUNDARIES OF SYSTOLIC
COMPLEXES

arXiv:1705.01062v1 [] 2 May 2017

TOMASZ PRYTULA
Abstract. Given a group G acting geometrically on a systolic complex X and
a hyperbolic isometry h ∈ G, we study the associated action of h on the systolic boundary ∂X. We show that h has a canonical pair of ﬁxed points on the
boundary and that it acts trivially on the boundary if and only if it is virtually
central. The key tool that we use to study the action of h on ∂X is the notion
of a K–displacement set of h, which generalises the classical minimal displacement
set of h. We also prove that systolic complexes equipped with a geometric action
of a group are almost extendable.

1. Introduction
A systolic complex is a simply connected simplicial complex whose vertex links
satisfy a certain combinatorial condition called 6–largeness. The condition of 6–
largeness serves as an upper bound for the combinatorial curvature, and thus systolic
complexes may be seen as combinatorial analogues of metric spaces of nonpositive
curvature, the so-called CAT(0) spaces. Systolic complexes were first introduced in
[Che00] under the name of bridged complexes, although their 1–skeleta had appeared
much earlier in metric graph theory (see e.g., [SC83]). In this article we are interested
in systolic complexes that are equipped with a geometric action of a group. Any such
group is called a systolic group. The theory of systolic complexes and groups, as
developed in [JŚ06], is to a large extent parallel to the theory of CAT(0) spaces and
groups. In particular, over the last fifteen years many of the nonpositive curvaturelike properties of systolic complexes have been established (see [Che00, JŚ06, Els09,
Els09b, Els09c, OP16] and references therein). On the other hand, a combinatorial
approach led to constructions of examples of systolic groups whose behaviour is very
different from the classical nonpositively curved groups [JŚ06].
An important invariant of a CAT(0) space X is its boundary at infinity ∂X. The
boundary is a topological space which, as a set, consists of equivalence classes of
geodesic rays in X, such that asymptotic rays are equivalent. One topologises it in a
Date: May 3, 2017.
2010 Mathematics Subject Classification. 20F67 (Primary), 20F65, 20F69 (Secondary).
Key words and phrases. Systolic complex, boundary at inﬁnity, hyperbolic isometry.
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way that two geodesic rays are ‘close’ if they fellow travel ‘long time’. Any G–action
by isometries on X gives rise to a G–action by homeomorphisms on ∂X. It turns out
that many algebraic properties of a group are reflected in topological properties of
the boundary and in the action itself.
In this article we study this correspondence in the setting of systolic complexes.
The boundary for systolic complexes was constructed in [OP09]. The construction is
similar to the one for CAT(0) spaces, however it is much more technical. The points
of the systolic boundary are also represented by geodesic rays in (the 1–skeleton of)
a systolic complex X, but not every geodesic ray in X gives a point in the boundary:
in order to ensure good properties of the boundary, a choice of a certain subclass of
geodesics was necessary. This is mainly due to the fact that arbitrary geodesics in
a systolic complex do not satisfy any form of the Fellow Traveller Property (indeed,
two geodesics of length D with the same endpoints may get D
2 apart). In [OP09]
the authors introduce good geodesics and good geodesic rays, and define the systolic
boundary ∂X as a set of equivalence classes of good geodesic rays in X. The topology
on ∂X is defined analogously as in the CAT(0) case. Both good geodesics and good
geodesic rays are preserved by simplicial automorphisms of X, and therefore any simplicial G–action on X induces a G–action (by homeomorphisms) on ∂X. Intuitively,
a good geodesic ray is a geodesic ray which, whenever contained in a flat F , follows
the CAT(0) geodesic in F . In particular good geodesic rays have the desired metric
properties, similar to those of geodesic rays in CAT(0) spaces.
An isometry (i.e., a simplicial automorphism) h of a systolic complex X is hyperbolic
if it does not fix any simplex of X. Note that if G acts geometrically on X then every
infinite order element of G is a hyperbolic isometry of X. The main point of this
article is to study the associated action of h on the systolic boundary ∂X. We start
by determining when this action is trivial (i.e., when h acts as the identity on ∂X).
Denote by CG (h) the centraliser of h in G. The following is the systolic analogue of
a result of K. Ruane for CAT(0) spaces [Rua01].
Theorem 1 (Theorem 4.5). Let G be a group acting geometrically on a systolic
complex X, and let h ∈ G be a hyperbolic isometry. Then h acts trivially on the
boundary ∂X if and only if the centraliser CG (h) has finite index in G.
The canonical object used to study the action of h on X is the minimal displacement
set of h, which is a subcomplex of X spanned by all the vertices which are moved
by h the minimal (combinatorial) distance. This distance is called the translation
length of h and it is denoted by L(h). Due to a ‘coarse nature’ of ∂X, in order to
study the action of h on ∂X it is convenient to replace the minimal displacement set
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by its coarse equivalent – the K–displacement set of h, for some K > L(h), which is
a subcomplex of X spanned by all the vertices that are moved by h the distance at
most K. The K–displacement set has all the desired (from our point of view) features
of the minimal displacement set, while it has the advantage of being more flexible as
one can let K vary.
The proof of Theorem 1 is based on the interplay between K–displacement sets of
h (for different values of K) and the centraliser CG (h). In particular, the ‘if’ direction
essentially boils down to showing the following two facts:
(1) Any point in ∂X represented by a geodesic ray that lies inside some K–displacement set of h is fixed by h.
(2) The centraliser CG (h) acts cocompactly on any K–displacement set of h.
The ‘only if’ direction is more involved. In this case we are given the information
about the action on the boundary, and we need to extract the information about the
action on the complex. For this we need X to satisfy the following property. We say
that X is almost extendable if there exists a constant E > 0 such that for every pair
of vertices x, y in X there is a good geodesic ray issuing from x and passing within
distance E from y. The following theorem is also of independent interest.
Theorem 2 (Theorem 3.2). Let X be a noncompact systolic complex, on which a
group G acts geometrically. Then X is almost extendable.
The proof of this theorem relies on the study of topology at infinity of systolic
complexes. It is similar to the proof of an analogous theorem for CAT(0) spaces
[Ont05]. The main difference is that our proof uses the notion of connectedness at
infinity, whereas the one in [Ont05] uses cohomology with compact supports. The
key fact is that in the setting above, the complex X is not 1–connected at infinity
(see [Osa07]).
In the second part of the article we consider arbitrary hyperbolic isometries of
X (not necessarily the virtually central ones). One can still ask whether such an
isometry h has any fixed points in ∂X. In the setting of CAT(0) spaces, a hyperbolic
isometry h has an axis, that is, an h–invariant geodesic line, and this axis determines
two fixed points of h in ∂X. In our situation h also has a kind of axis (see [Els09b]),
but unfortunately this axis does not have to determine an h–invariant good geodesic.
In fact, an h–invariant good geodesic may not exist. However, we do prove that h
has a pair of fixed points in ∂X.

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Theorem 3 (Proposition 6.2). Let G be a group acting geometrically on a systolic
complex X and let h ∈ G be a hyperbolic isometry. Then:

(1) there exist points h−∞ and h+∞ in the boundary ∂X which are fixed by h,
(2) for any vertex x ∈ X we have (hn · x)n → h+∞ and (h−n · x)n → h−∞ as
n → ∞ in the compactification X = X ∪ ∂X.

The second statement shows that h+∞ and h−∞ are in a certain sense the canonical
fixed points of h. To find h+∞ and h−∞ we construct an ‘almost h–invariant’ good
geodesic in X, by which we mean a geodesic that is contained in some K–displacement
set of h. This requires analysing the construction of good geodesics. We go through
the steps of the construction and show that given any two vertices x and y in the
minimal displacement set of h, a good geodesic between x and y is contained in a
K–displacement set where K is independent of distance between x and y. Then we
construct a bi-infinite good geodesic as a limit of finite good geodesics contained in
the K–displacement set.
In order to prove the second part of Theorem 3 we also study good geodesics
contained in the flats of X. In particular, we give a simple criterion for a geodesic
contained in a flat to be a good geodesic and we show that any geodesic that is good
in the flat is also good in the complex X.
We believe that the results presented in this article may be used in the further
study of systolic groups via their boundaries. Theorems 1 and 3 are the first steps in
analysing the dynamics of the action of h on ∂X, which in the CAT(0) setting plays
the key role in e.g., [PS09], where the topology of ∂X is related to splittings of G over
2–ended subgroups. Theorem 2 seems to be of a more general nature; its CAT(0)
counterpart has been widely used in the study of CAT(0) groups and boundaries.
Organisation. The article consists of an introductory Section 2, where we give background on systolic complexes and boundaries, and of the two main parts. In the first
part, which occupies Sections 3 and 4, we prove Theorem 2 and after establishing
basic facts about K–displacement sets we give a proof of Theorem 1. In the second
part (Sections 5 and 6) we first sketch the construction of good geodesics, and then
we prove Theorem 3.
Acknowledgements. I would like to thank Piotr Przytycki for suggesting the problem
and for many helpful discussions. I would also like to thank Damian Osajda for
helpful discussions and comments, in particular leading to the proof of Theorem 2.
Finally, I thank Dieter Degrijse and Damian Osajda for a careful proofreading of the
manuscript and help in improving the exposition.
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HYPERBOLIC ISOMETRIES AND BOUNDARIES

The author was supported by the Danish National Research Foundation through
the Centre for Symmetry and Deformation (DNRF92) and by the EPSRC First Grant
EP/N033787/1.
2. Systolic complexes and their boundaries
In this section we give some background on systolic complexes and their boundaries.
We also fix the terminology and notation that is used throughout the article.
2.1. Systolic simplicial complexes. Let X be a simplicial complex. We assume
that X is finite dimensional and uniformly locally finite, i.e., there is a uniform bound
on the degree of vertices in X. We equip X with the CW–topology, and always treat
it as a topological space (we do not make a distinction between an abstract simplicial
complex and its geometric realisation). Let X (k) denote the k–skeleton of X. In
particular X (0) is the vertex set of X. For any subset A ⊂ X (0) , a subcomplex
spanned by A is the largest subcomplex of X that has A as its vertex set. We denote
this subcomplex by span(A). A map f : X → Y of simplicial complexes is simplicial
if f (X (0) ) ⊂ Y (0) and whenever vertices x0 , x1 , . . . , xn span a simplex of X then their
images f (x0 ), f (x1 ), . . . , f (xn ) span a simplex of Y . Note that a simplicial map is
continuous, and in particular a simplicial automorphism is a homeomorphism of X.
In this article we will be particularly interested in metric aspects of simplicial
complexes.
Definition 2.1. We endow the vertex set X (0) with a metric, where the distance
d(x, y) between vertices x and y is defined to be the combinatorial distance in the
1–skeleton, i.e., the minimal number of edges of an edge–path joining x and y.
For two subcomplexes A, B ⊂ X, we define the distance d(A, B) to be the minimal
distance between vertices a ∈ A and b ∈ B.
Whenever we refer to the metric on X we mean the metric on X (0) defined above.
Consequently, a geodesic in X is a sequence of vertices (v0 , v1 , . . . , vn ) such that for
any 0 6 i, j 6 n we have d(vi , vj ) = |j − i|. Analogously we define a geodesic that
is infinite in one or both ends. In the first case we call it a geodesic ray, in the
second case: a geodesic line or a bi-infinite geodesic. Observe that a simplicial map
is 1–Lipschitz and any simplicial automorphism is an isometry of X.
We now briefly recall the notions needed to define systolic complexes. We say that
X is flag if every set of vertices of X pairwise connected by edges spans a simplex
of X. A flag simplicial complex is completely determined by its 1–skeleton X (1) or,
equivalently, by its vertex set X (0) and the metric d defined above. For a vertex
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v ∈ X the link of v is a subcomplex Lk(v, X) of X, that consists of all the simplices
of X that do not contain v, but together with v span a simplex of X. A cycle in X is
the image of a simplicial map f : S 1 → X from the triangulation of the 1–sphere to
X. A cycle is embedded if f is injective. Let γ be an embedded cycle. The length of
γ, denoted by |γ| is the number of edges of γ. A diagonal of γ is an edge in X that
connects two nonconsecutive vertices of γ.
We are ready to define systolic complexes. Our main reference for the theory of
systolic complexes is [JŚ06].
Definition 2.2. Given a natural number k > 6, a simplicial complex X is k–large
if every embedded cycle γ in X with 4 6 |γ| < k has a diagonal. We say that X is
∞–large if it is k–large for every k > 6.
Definition 2.3. A simplicial complex X is k–systolic if it is simply connected and
if for every vertex v ∈ X the link Lk(v, X) is flag and k–large. If k = 6 then we
abbreviate 6–systolic to systolic.
A k–systolic complex is flag and k–large [JŚ06, Proposition 1.4 and Fact 1.2(4)].
Note that if k 6 m then ‘m–systolic’ implies ‘k–systolic’. In this article we will be
interested in the (most general) case of k = 6. This case is of particular importance
in the theory, as for k > 7 one shows that k–systolic complexes are δ–hyperbolic
[JŚ06, Theorem 2.1].
The condition of k–largeness, when applied to the link of a vertex v ∈ X, serves
as a kind of upper bound for the curvature around v. In particular complexes with
6–large links are called complexes of simplicial nonpositive curvature (SNPC). Consequently, systolic complexes can be thought of as simplicial analogues of CAT(0)
metric spaces.
Definition 2.4. Let v ∈ X be a vertex and let n be a positive integer. Define the
ball of radius n centred at v by Bn (v, X) = span{x ∈ X (0) | d(x, v) 6 n} ⊂ X. Define
the sphere of radius n centred at v by Sn (v, X) = span{x ∈ X (0) | d(x, v) = n}. For
a subcomplex A ⊂ X define the ball of radius n around A by
[
Bn (A, X) =
Bn (v, X).
v∈A(0)

We also refer to Bn (A, X) as an n–neighbourhood of A in X.
A subcomplex A ⊂ X is convex if for every two vertices x, y ∈ A any geodesic
between x and y in X is contained in A. Note that since geodesics in X are not
necessarily unique, a subcomplex A ⊂ X can be isometrically embedded and not
convex.
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Proposition 2.5. Let X be a systolic complex. Then the following hold:
(1) For any convex subcomplex A ⊂ X the ball Bn (A, X) is convex and contractible [JŚ06, Corollary 7.5]. In particular for any vertex v ∈ X the ball
Bn (v, X) is convex and contractible.
(2) The complex X is contractible [JŚ06, Theorem 4.1(1)].
We finish this section with some terminology regarding group actions on simplicial
complexes. Let G be a (discrete) group acting on a simplicial complex X. We assume
that G acts via simplicial automorphisms. We say that the action is:
• proper if for every vertex v ∈ X the stabiliser Gv is finite,
• cocompact if there exists a compact subset K ⊂ X such that G · K = X,
• geometric if it is proper and cocompact.
A group is called systolic if it acts geometrically on a systolic complex.
2.2. Boundaries of systolic complexes. Given a (noncompact) systolic complex
X one can define the boundary at infinity (or shortly the boundary) ∂X of X. Analogously to the cases of δ–hyperbolic and CAT(0) spaces, the boundary for systolic
complexes is given by a set of equivalence classes of geodesic rays, such that asymptotic rays are equivalent. In this section we give the definition of the boundary and
briefly discuss its key features that are needed in this article. For more details we
refer the reader to [OP09].
The main difference from δ–hyperbolic and CAT(0) cases is that, instead of arbitrary geodesic rays, to define the boundary one uses a canonically defined subcollection of the so-called good geodesic rays. To define good geodesic rays one first defines
good geodesics. The actual definition of good geodesics (which is quite involved) is
needed only in Section 6, and therefore we give this definition in Section 5.
In order to follow the arguments in Sections 3 and 4 it is enough to know that a
good geodesic is a certain geodesic in X, and that the subclass of good geodesics has
the following properties:
(1) for any two vertices there exists a (not necessarily unique) good geodesic
joining these vertices,
(2) any subgeodesic of a good geodesic is a good geodesic,
(3) any simplicial automorphism of X maps good geodesics to good geodesics.
A good geodesic ray is a geodesic ray, such that any of its finite subgeodesics is a
good geodesic. By (3) any simplicial automorphism of X maps good geodesic rays to
good geodesic rays.
Let R denote the set of all good geodesic rays in X. For a vertex O ∈ X let RO
denote the set of all good geodesic rays starting at O.
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Definition 2.6. Let X be a systolic complex. Define the boundary of X to be the
set ∂X = R/ ∼ where for rays η = (v0 , v1 , . . .) and ξ = (w0 , w1 , . . .) we have η ∼ ξ if
and only if there exists K > 0, such that for every i > 0 we have d(vi , wi ) 6 K.
Define the boundary of X with respect to the basepoint O to be the set ∂O X =
RO / ∼, where ∼ is the same equivalence relation as above. In both cases let [η]
denote the equivalence class of η.
For any vertex O ∈ X there is a bijection ∂X → ∂O X [OP09, Corollary 3.10].
In particular this means that for every geodesic ray η ⊂ X and for every vertex
O ∈ X there is a ray ξ ⊂ X starting at O such that [η] = [ξ] in ∂X. This fact will
be used many times in this article. The set X = X ∪ ∂O X can be equipped with
a topology that extends the standard topology on X, and turns X into a compact
topological space [OP09, Propositions 4.4 and 5.3]. For any two vertices O, O ′ ∈ X
there is a homeomorphism between X ∪ ∂O X and X ∪ ∂O′ X [OP09, Lemma 5.5]. Any
simplicial action of a group on X extends to an action by homeomorphisms on X
[OP09, Theorem A(4)].
In this article we will mostly be concerned with the induced action on the boundary,
not on the entire X. Moreover, we will be interested in the action on the boundary
seen as a set, not as a topological space. For this, we can use a slightly simpler
definition.
Definition 2.7. Suppose that a group G acts simplicially on X. We define an action
of G on the set ∂X as follows. Let [η] ∈ ∂X where η = (v0 , v1 , . . .). Then define
g · [η] = [g · η] where g · η = (g · v0 , g · v1 , . . .). It is straightforward to check that this
is well defined and it defines an action of G on ∂X.
One can also verify, that via the bijection ∂X → ∂O X the action described above
agrees with the action on ∂O X defined in [OP09].
We conclude this section with certain metric properties of good geodesics. The
following is a crucial property, which can be seen as a coarse version of CAT(0)
inequality for good geodesics.
Theorem 2.8. [OP09, Corollary 3.4] Let (v0 , v1 , v2 , . . . , vn ) and (w0 , w1 , w2 , . . . , wm )
be good geodesics in a systolic complex X such that v0 = w0 . Then for any 0 6 c 6 1
we have
d(v⌊cn⌋ , w⌊cm⌋ ) 6 c · d(vn , wm ) + D,
where D is a universal constant.
This leads to the following corollary, which will be very useful to us.
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Corollary 2.9. Let η = (v0 , v1 , v2 , . . .) and ξ = (w0 , w1 , w2 , . . .) be good geodesic rays
in a systolic complex X. If [η] = [ξ] in ∂X then for every i > 0 we have
d(vi , wi ) 6 d(v0 , w0 ) + 2D + 1,
where D is the constant appearing in Theorem 2.8.
Proof. Since [η] = [ξ], there is a constant K > 0 such that for all i we have d(vi , wi ) 6
K. Fix i > 0, pick n > K and let z = (z0 = v0 , z1 , z2 , . . . , wni ) be a good geodesic
joining v0 and wni . By Theorem 2.8 applied to (v0 , v1 , v2 , . . . , vni ) and z we have
d(vi , zi ) 6

K
1
d(vni , wni ) + D 6
+ D 6 1 + D.
n
n

(2.1)

Applying Theorem 2.8 to z and (w0 , w1 , w2 , . . . , wni ) (with the direction reversed)
we obtain that d(zi , wi ) 6 d(v0 , w0 ) + D. This, together with (2.1) and the triangle
inequality gives the claim.

3. Almost extendability of systolic complexes
In this section we study a property of metric spaces called the almost extendability.
This property can be defined for arbitrary geodesic metric spaces. The definition we
present is adjusted to the setting of systolic complexes.
Definition 3.1. A systolic complex X is almost extendable, if there exists a constant
E > 0 such that for any two vertices x and y of X, there is a good geodesic ray
starting at y and passing within distance E from x.
It is easy to construct systolic complexes (in fact, trees) that are not almost extendable. For example, let T denote the half-line R+ with the interval of length n
attached to every integer n ∈ R+ . The standard triangulation turns T into a systolic
complex in which every combinatorial geodesic is a good geodesic. One can easily see
that T is not almost extendable. When we equip a systolic complex with a geometric
action of a group then the situation changes.
Theorem 3.2. Let X be a noncompact systolic complex, on which a group G acts
geometrically. Then X is almost extendable.
The analogous theorem is true in the CAT(0) setting [Ont05, Theorem B], and it
is an exercise in the setting of δ–hyperbolic groups (see [Ont05]). Our proof is similar
to the one for CAT(0) spaces, however, it can be seen as more direct. The main
difference is that instead of cohomology with compact supports, our proof uses the
notion of connectedness at infinity. We begin by recalling this notion.
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Definition 3.3. Let Y be a topological space and let n > −1 be an integer. We
say that Y is n–connected at infinity if for every −1 6 k 6 n the following condition
holds: for every compact set K ⊂ Y there exists a compact set L ⊂ Y such that
K ⊂ L and every map S k = ∂B k+1 → Y \ L extends to a map B k+1 → Y \ K.
For k = −1 we define S −1 = ∅ and B 0 = {∗}. In particular Y is (−1)–connected
at infinity if and only if it is not compact.
Note that if Y is a simplicial complex then (in view of the Simplicial Approximation
Theorem) in the above definition it is enough to consider only simplicial maps.
The following theorem of D. Osajda is the crucial ingredient in the proof of Theorem 3.2.
Theorem 3.4. [Osa07, Theorem 3.2] Let X be a noncompact systolic complex, on
which a group G acts geometrically. Then X is not 1–connected at infinity.
Proof of Theorem 3.2. First we show that it is enough to prove the following claim.
Claim 1. Let p be a fixed vertex. Then there exists a constant E ′ such that, for any
g ∈ G there is a good geodesic ray starting at p and passing within E ′ from g · p.
Indeed, let x and y be arbitrary vertices of X. By cocompactness there exists
R > 0 and elements g1 , g2 ∈ G such that we have d(g1 · p, x) 6 R and d(g2 · p, y) 6 R.
By Claim 1 there exists a good geodesic ray η starting at p and passing within E ′
from g1−1 g2 · p. Then the ray g1 · η starts at g1 · p and passes within E ′ from g2 · p, and
hence it passes within E ′ + R from y. Now let ξ be a good geodesic ray starting at x
and such that [ξ] = [g1 ·η]. Write g1 ·η = (g1 ·p = v0 , v1 , . . .) and ξ = (x = w0 , w1 , . . .).
Then by Corollary 2.9 for every i > 0 we have
d(vi , wi ) 6 d(g1 · p, x) + 2D + 1 6 R + 2D + 1.
Since g1 ·η passes within E ′ +R from y, we have that ξ passes within E ′ +R+R+2D+1
from y. Therefore Claim 1 implies the theorem (with constant E = E ′ + R + R +
2D + 1).
The rest of the proof is devoted to proving Claim 1. We need a little preparation.
In what follows, for a good geodesic or a good geodesic ray η we will denote its vertices
by η(i), for i > 0, i.e., η = (η(0), η(1), η(2), . . .). In other words, the geodesic η may
be seen as a map N → X. We still treat η as a subset of X; the above notation is
introduced only to simplify the exposition.
For a good geodesic η let lη denote the supremum of natural numbers l, such that
η can be extended to a good geodesic on the interval [0, l] = {0, 1, . . . , l} ⊂ N. Note
that lη does not have to be attained, in that case we write lη = ∞. Observe that
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HYPERBOLIC ISOMETRIES AND BOUNDARIES

if lη < ∞ then there is an extension of η to the interval [0, lη ]. If lη = ∞ then by
the fact that X ∪ ∂X is compact, there is an extension of η to the interval [0, ∞),
i.e., the geodesic η can be extended to a good geodesic ray. For vertices x, y ∈ X let
[[x, y]] denote a good geodesic between these two vertices. Note that such a geodesic
in general is not unique.
Now we begin the proof of Claim 1. We proceed by contradiction. Assume that
Claim 1 does not hold, then we have the following:
(∗) For every r > 0 there exists gr ∈ G such that for every vertex x ∈ Br (gr · p, X)
we have l[[p,x]] < ∞ for every good geodesic [[p, x]].
Claim 2. For every r > 0 we have
sup{l[[p,x]] | [[p, x]] where x ∈ Br (gr · p, X)} < ∞.
(The supremum is taken over all possible good geodesics that start at p and end at
a vertex of Br (gr · p, X).)
To prove Claim 2, assume conversely that there exists a sequence of good geodesics
([[p, xi ]])i with xi ∈ Br (gr ·p, X), such that l[[p,xi ]] → ∞ as n → ∞. Let ηi denote a good
geodesic extending [[p, xi ]] to the interval [0, l[[p,xi ]] ] (we choose one for each i). Using
a diagonal argument, out of the sequence (ηi )i one constructs an infinite geodesic
ray ξ that issues from p, and such that for any interval [0, l] we have ξ [0,l] = ηi for
some i = il (cf. [OP09, Proposition 5.3]). In particular, the ray ξ intersects the ball
Br (gr · p, X), which contradicts (∗).
Claim 3. For every r > 0 there exists r ′ > r such that for every vertex y ∈ X \
Br′ (p, X), every good geodesic [[p, y]] misses the ball Br (gr · p, X), i.e., we have
[[p, y]] ∩ Br (gr · p, X) = ∅.
Note that we have p ∈
/ Br (gr · p, X), for otherwise we would get a contradiction
with (∗) as there always is a geodesic ray issuing from p (since X is noncompact).
Let r ′ = sup{l[[p,x]] | [[p, x]] where x ∈ Br (gr · p, X)}. Then the claim follows from the
definition of r ′ .
Claim 4. The complex X is 1–connected at infinity.
First observe that since X is noncompact, it is (−1)–connected at infinity. Let
K ⊂ X be a compact subset. Take M > 0 such that K ⊂ BM (p, X) and consider the
ball BM +D+2 (p, X), where D is the constant appearing in Theorem 2.8 (the reason
why we need to pass to the larger ball will become clear later on).
Pick r > M + D + 2. By Claim 3 (after ‘translating its statement by gr−1 ’) there
exists r ′ > r such that every good geodesic joining a vertex y ∈ X \ Br′ (gr−1 · p, X)
11

TOMASZ PRYTULA

with gr−1 · p, misses the ball Br (p, X). Set L = Br′ (gr−1 · p, X). By construction we
have K ⊂ L, and gr−1 · p ∈ L \ K.
Suppose f : S 0 → X \ L is a simplicial map. Let v1 and v2 be the two vertices in
the image of f . For i ∈ {0, 1} let ηi be a good geodesic joining gr−1 · p with vi . Both
ηi miss the ball Br (p, X) (and hence they miss K) and therefore their union defines
a map F : B 1 → X \ K that extends f . This shows that X is 0–connected at infinity.
Now let f : S 1 → X \ L be a simplicial map. Let (v0 , v1 , . . . , vn , vn+1 = v0 ) be the
vertices of the image of f appearing in this order, i.e., for all i vertices vi and vi+1
are adjacent. For every i ∈ {0, . . . , n} let ηi be a good geodesic joining gr−1 · p and
vi . Observe that no ηi intersects the ball Br (p, X). We will use ηi ’s to construct the
required extension of f to the disk B 2 .
For any i consider the cycle αi ⊂ X which is the union
αi = ηi ∪ ηi+1 ∪ [vi , vi+1 ].
We will show that αi can be contracted to a point in its (D + 2)–neighbourhood.
First note that either ηi and ηi+1 have the same length, or their lengths differ by 1.
In the first case put k = d(gr−1 · p, vi ) = d(gr−1 · p, vi+1 ). Since ηi and ηi+1 start at
the same vertex and end at vertices that are adjacent, it follows from Theorem 2.8
that for any j ∈ {0, 1, . . . , k} we have
d(ηi (j), ηi+1 (j)) 6 D + 1.
For any j ∈ {0, . . . , k} let βji be a geodesic between ηi (j) and ηi+1 (j) (note that β0i
is the vertex gr−1 · p and βki is the edge [vi , vi+1 ]). Now for every j ∈ {0, . . . , k − 1}
consider a cycle γji defined as
i
γji = βji ∪ [ηi+1 (j), ηi+1 (j + 1)] ∪ βj+1
∪ [ηi (j), ηi (j + 1)].

By construction γji is contained in the ball BD+2 (ηi (j), X) and therefore it can be contracted inside BD+2 (ηi (j), X), as balls in X are contractible (see Proposition 2.5.(1)).
These contractions of γji for all j ∈ {0, . . . , k} form a contraction of αi inside the ball
BD+2 (ηi , X) around the geodesic ηi . (Formally, by a contraction we mean a simplicial map from a simplicial disk f : B 2 → BD+2 (ηi , X) such that f maps the boundary
∂B 2 isomorphically onto αi .)
In the second case, assume that ηi+1 is longer than ηi , i.e., we have d(gr−1 ·p, vi ) = k
and d(gr−1 · p, vi+1 ) = k + 1. In this case the concatenation ηi ∪ [vi , vi+1 ] is a geodesic.
Then it follows from [JŚ06, Lemma 7.7] that ηi+1 (k) and vi are adjacent, and therefore
ηi+1 (k), vi and vi+1 span a 2–simplex. Now ηi and ηi+1 [0,k] are of the same length
and their endpoints vi and ηi+1 (k) are adjacent. Proceeding as in the first case we
12

HYPERBOLIC ISOMETRIES AND BOUNDARIES

obtain a contraction of the cycle
ηi ∪ ηi+1

[0,k]

∪ [vi , ηi+1 (k)]

inside the ball BD+2 (ηi , X). Adding the 2–simplex [vi , ηi+1 (k), vi+1 ] we obtain the
desired contraction of αi = ηi ∪ ηi+1 ∪ [vi , vi+1 ].
Finally, contractions of αi for all i ∈ {0, . . . , n} form the contraction of (v0 , . . . vn , v0 )
that is performed in the (D + 2)–neighbourhood of the union of all ηi ’s. Since every ηi
misses the ball Br (p, X), the (D + 2)–neighbourhood of ηi misses the ball BM (p, X)
as r > M + D + 2, and hence it misses K as K ⊂ BM (p, X). We conclude that the
constructed contraction of (v0 , . . . , vn , v0 ) defines the extension of f that misses K.
This finishes the proof of Claim 4.
This gives a contradiction with Theorem 3.4 and hence proves Claim 1.



4. Isometries acting trivially on the boundary
In this section, given a group G acting geometrically on a systolic complex X, we
investigate which elements of G act trivially on the boundary ∂X. Before proving
the main theorem which characterises such elements in terms of their centralisers in
G, we introduce the terminology and briefly discuss the tools needed in the proof.
4.1. Hyperbolic isometries and their K–displacement sets. Let h be an isometry (i.e., a simplicial automorphism) of a systolic complex X. We say that h is
hyperbolic if it does not fix any simplex of X. If h is hyperbolic, then any of its
powers is hyperbolic as well ([Els09b]). To such h one associates the displacement
function dh : X (0) → N defined as dh (x) = d(x, h · x). The minimum of dh (which is
always attained) is called the translation length of h and is denoted by L(h).
Definition 4.1. Let h be a hyperbolic isometry of a systolic complex X. The minimal
displacement set Min(h) is the subcomplex of X spanned by all the vertices of X which
are moved by h the minimal distance, i.e.:
Min(h) = span{x ∈ X (0) | d(x, h · x) = L(h)}.
More generally, for a natural number K > L(h) define the K–displacement set as
DispK (h) = span{x ∈ X (0) | d(x, h · x) 6 K}.
Clearly we have DispK (h) ⊂ DispK ′ (h) for K 6 K ′ and DispL(h) (h) = Min(h).
Let us mention that Min(h) is a systolic complex on its own, and its inclusion into
X is an isometric embedding ([Els09b]). We do not know whether the same is true for
13

TOMASZ PRYTULA

DispK (h) for K > L(h). In this article we are interested only in the coarse-geometric
behaviour of DispK (h).
Observe that if x ∈ X is a vertex such that d(x, DispK (h)) 6 C for some C > 0,
then by the triangle inequality we have d(x, h · x) 6 K + 2C. This means that
BC (DispK (h), X) ⊆ DispK+2C (h). In the presence of a geometric action of a group,
the (partial) converse also holds.
Lemma 4.2. Let G be a group acting geometrically on a systolic complex X and
suppose that h ∈ G is a hyperbolic isometry. Pick K > L(h). Then there exists
C > 0 such that for all K ′ 6 K we have DispK (h) ⊂ BC (DispK ′ (h), X).
The lemma is an easy consequence of the following theorem.
Theorem 4.3. Let G act geometrically on a systolic complex X, and let h ∈ G
be a hyperbolic isometry. Then for any K > L(h) the centraliser CG (h) ⊂ G acts
geometrically on the subcomplex DispK (h) ⊂ X.
Proof. The proof is a verbatim translation of K. Ruane’s proof of a similar result for
CAT(0) spaces [Rua01, Theorem 3.2]. The original proof treats only the case where
DispK (h) = Min(h), however it is straightforward to check that it carries through for
any DispK (h). We include the proof for the sake of completeness.
First we check that CG (h) leaves DispK (h) invariant. Let x ∈ DispK (h) be a vertex
and take g ∈ CG (h). Then we have
d(g · x, hg · x) = d(g · x, gh · x) = d(x, h · x) 6 K
and thus g · x ∈ DispK (h). Observe that the action of CG (h) on DispK (h) is proper,
since the action of G on X is proper. We only need to check cocompactness. We
proceed by contradiction. Assume that there is no compact subset of DispK (h) whose
CG (h)–translates cover DispK (h), and pick a vertex x0 ∈ DispK (h). Then there exists
a sequence of vertices (xn )∞
n=1 of DispK (h) such that d(CG (h)·x0 , xn ) → ∞ as n → ∞.
Let D ⊂ X be a compact set containing x0 such that G · D = X, and let (gn )∞
n=1 be
a sequence of elements of G such that gn · xn ∈ D. We can assume (by passing to a
subsequence if necessary) that gn 6= gm for n 6= m. Indeed, we have
d(x0 , gn−1 · x0 ) > d(x0 , xn ) − diamD → ∞ as n → ∞.
Now consider the family of elements {gn hgn−1 }n>1 of G. We claim that the displacement functions dgn hgn−1 are uniformly bounded on D. Let y ∈ D be a vertex.
14

HYPERBOLIC ISOMETRIES AND BOUNDARIES

We have
dgn hgn−1 (y) = d(y, gn hgn−1 · y)

6 d(y, gn · xn ) + d(gn · xn , gn hgn−1 (gn · xn ))
+ d(gn hgn−1 (gn · xn ), gn hgn−1 · y)

= d(y, gn · xn ) + d(gn · xn , gn h · xn ) + d(gn h · xn , gn hgn−1 · y)
6 diam(D) + K + diam(D).
−1 for n 6= m (after passing to
Since G acts properly, it must be gn hgn−1 = gm hgm
−1 g ∈ C (h). Now for any
a subsequence). Therefore for all n 6= m we have that gm
n
G
n 6= 1 we get

d(xn , gn−1 g1 · x0 ) 6 d(xn , gn−1 g1 · x1 ) + d(gn−1 g1 · x1 , gn−1 g1 · x0 ) 6 diam(D) + d(x0 , x1 ).
This gives a contradiction since gn−1 g1 ∈ CG (h), and by the choice of xn we have
that d(xn , CG (h) · x0 ) → ∞ as n → ∞.

We are ready now to prove Lemma 4.2.
Proof of Lemma 4.2. Pick any K ′ 6 K and let x0 ∈ DispK ′ (h) be a vertex. By
Theorem 4.3 the centraliser CG (h) acts cocompactly on DispK (h). Hence there exists
R > 0 such that DispK (h) ⊂ CG (h) · BR (x0 , X). Since CG (h) · x0 ⊂ DispK ′ (h), taking
R as C proves the lemma.

Remark 4.4. We believe that in Lemma 4.2 one can obtain a concrete distance estimate, i.e., in the formula DispK ′ (h) ⊂ BC (DispK (h), X) one can express C as an
explicit function of K and K ′ . However, for our purposes, the existence of any constant C is sufficient.
4.2. Trivial action on the boundary. In this section we characterise hyperbolic
isometries that act trivially on the boundary as being virtually central. More precisely,
we show the following.
Theorem 4.5. Let G be a group acting geometrically on a systolic complex X, and
let h ∈ G be a hyperbolic isometry. Then h acts trivially on the boundary ∂X if and
only if the centraliser CG (h) has finite index in G.
The theorem is a systolic analogue of a theorem of K. Ruane for CAT(0) spaces
[Rua01, Theorem 3.4]. In a certain way, our situation is more restrictive. Namely,
by [OP16, Corollary 5.8] the centraliser CG (h) is commensurable with the product
Fn × Z, where Fn is the free group on n generators for some n > 0. It follows that
either of the assertions of Theorem 4.5 holds true if and only if the group G itself is
commensurable with Fn × Z.
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TOMASZ PRYTULA

Proof of Theorem 4.5. “if ” direction. By Theorem 4.3 the centraliser CG (h) acts
cocompactly on the minimal displacement set Min(h) ⊂ X. Since the index [G : CG (h)]
is finite, it follows that the action of CG (h) on X is cocompact as well. Therefore
there exists a constant K > 0 such that for any vertex x ∈ X, there is a vertex
y ∈ Min(h) with d(x, y) 6 K. Hence, by the triangle inequality, for any x ∈ X we
have d(x, h · x) 6 L(h) + 2K.
Now let η = (v0 , v1 , v2 , . . .) be a good geodesic ray in X. For any i > 0 we have
d(vi , h · vi ) 6 L(h) + 2K and hence [η] = [h · η] in ∂X.
“only if ” direction. Choose a vertex y ∈ X and let x ∈ X be an arbitrary
vertex. By Theorem 3.2 there exists a good geodesic ray η = (v0 , v1 , v2 , . . .) such that
v0 = y and for some i > 0 we have d(vi , x) 6 E, where E is a constant independent
of x and y.
The isometry h acts trivially on the boundary, so we have [η] = [h · η]. Applying
Corollary 2.9 we obtain
d(vi , h · vi ) 6 d(v0 , h · v0 ) + 2D + 1,
where D is the constant appearing in Theorem 2.8. In other words, we have vi ∈
DispK (h) where K = d(v0 , h · v0 ) + 2D + 1. Since d(vi , x) 6 E, the triangle inequality
implies that x ∈ DispK+2E (h) (see the discussion after Definition 4.1). Because x was
arbitrary, we have X = DispK+2E (h). By Theorem 4.3 the centraliser CG (h) acts
cocompactly on X and so it has finite index in G.

We obtain the following corollary.
Corollary 4.6. Let G be a torsion-free group, acting geometrically on a systolic
complex X. Then G acts trivially on ∂X if and only if G ∼
= Z or G ∼
= Z2 .
Proof. If G is isomorphic to either Z or Z2 then every element of G is central, and by
Theorem 4.5 it acts trivially on ∂X.
Now assume that G acts trivially on ∂X. Since G is torsion-free, all of its elements
are hyperbolic, and therefore by Theorem 4.5 every element is virtually central. Pick
h ∈ G. By [OP16, Corollary 5.8] the centraliser CG (h) contains a finite-index subgroup H ∼
= Fn × Z, where Fn is the free group on n generators for some n > 0. We
must have n 6 1 for otherwise no non-trivial element of Fn would be virtually central
in G. This means that H is isomorphic to either Z or Z2 . Now since H has finite
index in CG (h) and CG (h) has finite index in G we get that H has finite index in G.
If H ∼
= Z then G is a virtually cyclic torsion-free group, and hence it must be
isomorphic to Z. If H ∼
= Z2 then G is a torsion-free group that contains Z2 as a
finite-index subgroup. Such G must be isomorphic to either Z2 or to the fundamental
16

HYPERBOLIC ISOMETRIES AND BOUNDARIES

group of the Klein bottle. Since the latter contains elements that are not virtually
central, we conclude that G ∼

= Z2 .
5. Good geodesics
The proofs in Section 6 require going through the construction of good geodesics
(unlike proofs in previous sections, where only certain ‘formal’ properties were needed).
In this section we give a sketch of this construction.
In order to define good geodesics we first describe the construction of Euclidean
geodesics. This construction is fairly involved, and hence it is divided into a few
steps. Our exposition is based on [OP09, Sections 7–9] (we refer the reader there
for the proofs of various statements discussed below). Throughout this section let X
be a systolic complex. Some notions appearing in the construction are presented in
Figures 1 and 2 in Section 6 (in the special case of X = E2∆ ).
5.1. Directed geodesics. Let x, y ∈ X be two vertices and put n = d(x, y). A
directed geodesic from x to y is a sequence of simplices (σi )ni=0 such that σ0 = x,
σn = y and the following two conditions are satisfied:
(1) any two consecutive simplices σi and σi+1 are disjoint and together they span
a simplex of X,
(2) for any three consecutive simplices σi−1 , σi , σi+1 we have
Res(σi−1 , X) ∩ B1 (σi+1 , X) = σi ,
where Res(σi−1 , X) is the union of all simplices of X that contain σi−1 .
A directed geodesic from x to y always exists and it is unique. One can show that
σi ⊂ Si (x, X) ∩ Sn−i (y, X), and therefore any sequence of vertices (vi )ni=0 such that
vi ∈ σi is a geodesic. Finally, as the name suggests, directed geodesics in general are
not symmetric – usually a directed geodesic from x to y is not equal to a directed
geodesic from y to x.
5.2. Layers. The intersection Si (x, X) ∩ Sn−i (y, X) is called the layer i between x
and y and it is denoted by Li . For any i the layer Li is convex and ∞–large. (Layers
in fact can be defined in the same way for any two convex subcomplexes V and W
such that for every v ∈ V and w ∈ W one has d(v, w) = n for some fixed n > 0.)
Convention 5.1. Suppose that (σi )ni=0 is a directed geodesic from x to y and (τi )ni=0
is a directed geodesic from y to x. We introduce the following convention: despite
(σi )ni=0 and (τi )ni=0 go in the opposite directions, we index simplices of (τi )ni=0 in the
same direction as for (σi )ni=0 , i.e., τ0 = x, τ1 , . . . , τn−1 , τn = y.
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TOMASZ PRYTULA

Observe that both σi and τi are contained in the layer Li . Define the thickness
of layer Li (with respect to (σi )ni=0 and (τi )ni=0 ) to be the maximal distance between
vertices of σi and τi (since layers are convex, this distance is always realised inside
Li ). The layer is thin if its thickness is at most 1, and it is thick otherwise.
A pair of indices (j, k) such that 0 < j < k < n and j < k − 1 is called a thick
interval if layers Lj and Lk are thin, and for every i such that j < i < k the layer Li is
thick. If for some i we have j < i < k then we say that i belongs to the interval (j, k).
5.3. Characteristic surfaces. Let (j, k) be a thick interval and let si ∈ σi and
ti ∈ τi be vertices such that for any j 6 i 6 k the distance between si and ti is equal
to the thickness of layer Li . Consider the sequence of vertices
(sj , sj+1 , . . . , sk−1 , sk , tk , tk−1 , . . . , tj+1 , tj , sj ).
Observe that any two consecutive vertices in the above sequence are adjacent and
therefore this sequence defines a closed loop which we denote by γ. In fact γ is
always an embedded loop (this amounts to saying that sj 6= tj and sk 6= tk ).
Let S : ∆ → X be a minimal surface spanned by γ, i.e., a simplicial map from a
triangulation of a 2–disk ∆ such that:
(1) the boundary of ∆ is mapped isomorphically to γ,
(2) the disk ∆ consists of the least possible number of triangles (among all disks
∆′ for which there exists a simplicial map S ′ : ∆′ → X satisfying (1)).
We call S : ∆ → X a characteristic surface (for the thick interval (j, k)) and we call
∆ a characteristic disk. It is a standard fact that a minimal disk is always systolic,
i.e., every of its internal vertices is incident to at least 6 triangles.
The cycle γ does not have to be unique, and hence there could be many characteristic surfaces. For any two characteristic surfaces S : ∆ → X and S ′ : ∆′ → X
the disks ∆ and ∆′ are isomorphic. We can thus identify all such disks and denote
the characteristic disk by ∆. Now for any simplex ρ ∈ ∆ the images S(ρ) for all
possible characteristic surfaces span a simplex of X, which we denote S(ρ). This assignment, called the characteristic mapping, respects inclusions, i.e., if ρ1 ⊆ ρ2 then
S(ρ1 ) ⊆ S(ρ2 ).
5.4. Geometry of characteristic disks. For any i such that j 6 i 6 k, let vi and
wi be vertices of ∆ that are preimages of si and ti respectively, for some characteristic
surface S : ∆ → X. In fact vertices vi and wi are uniquely defined and the sequence
(vj , vj+1 , . . . , vk−1 , vk , wk , wk−1 , . . . , wj+1 , wj , vj )
constitutes the boundary of the disk ∆.
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HYPERBOLIC ISOMETRIES AND BOUNDARIES

Denote by E2∆ the equilateral triangulation of the Euclidean plane. Clearly E2∆
viewed as a simplicial complex is systolic. One shows that the characteristic disk
∆ can be isometrically embedded in E2∆ (such disk is called flat). Moreover, after
embedding ∆ ⊂ E2∆ , the edges [vj , wj ] and [vk , wk ] are parallel, and consecutive layers
in ∆ between them are contained in straight lines of E2∆ (treated as subcomplexes of
E2∆ ), that are parallel to the lines containing [vj , wj ] and [vk , wk ]. In particular for
any i the vertices vi and wi lie on a straight line inside E2∆ . The subpath of this line
between vi and wi is the unique geodesic between vi and wi in ∆, which we denote
by vi wi . The geodesic vi wi is in fact equal to the entire layer i in ∆ (between the
edges [vj , wj ] and [vk , wk ]).
Finally, for any characteristic surface S : ∆ → X (and hence for a characteristic
mapping S : ∆ → X as well) the image of the geodesic vi wi is contained in the layer
i in X. Also, any characteristic surface S : ∆ → X restricted to vi wi is an isometric
embedding.
5.5. Euclidean diagonals. Given the characteristic disk ∆, for every i ∈ {j, . . . , k}
let vi′ and wi′ be points on the unique geodesic between vi and wi in ∆, that are at
distance 12 from vi and wi respectively. In particular vj′ = wj′ and vk′ = wk′ . Consider
a piecewise linear loop defined as the concatenation of straight segments between
consecutive points in the sequence
′
′
′
′
, . . . , wj+1
, wj′ = vj′ ),
, vk′ = wk′ , wk−1
(vj′ = wj′ , vj+1
, . . . vk−1

and let ∆′ be a polygonal domain inside ∆ enclosed by this loop. We call ∆′ the
modified characteristic disk. We endow ∆′ with a path metric induced from the
Euclidean metric on E2∆ ∼
= E2 . Observe that ∆′ is simply connected, and therefore
this path metric is in fact a CAT(0) metric. (The disk ∆′ does not have to be a
convex subset of E2∆ with respect to the Euclidean metric on E2∆ , and therefore a
CAT(0) geodesic inside ∆′ does not have to be a straight line.)
k−1
The Euclidean diagonal of ∆ is a sequence of simplices (ρi )j+1
of ∆ defined as
′
′
′
follows. Let α be a CAT(0) geodesic in ∆ between points vj = wj and vk′ = wk′ . For
every i such that j < i < k choose vertices on the geodesic vi wi , that are closest to
the point of intersection α ∩ vi wi . For any i, it is either a single vertex, in that case
we put ρi to be that vertex, or in the case when α goes through the barycentre of
some edge of vi wi , then we put ρi to be this edge. One can show that the Euclidean
diagonal for ∆ satisfies the following two conditions:
(1) for any i such that j < i < k − 1 simplices ρi and ρi+1 span a simplex,
(2) vertices vj , wj , σj+1 span a simplex and vertices vk , wk , ρk−1 span a simplex
(in particular ρj+1 and ρk−1 are necessarily vertices).
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TOMASZ PRYTULA

5.6. Euclidean geodesics. We are ready now to define Euclidean geodesics.
Definition 5.2. The Euclidean geodesic between vertices x and y in X, such that
d(x, y) = n is the sequence of simplices (δi )ni=0 defined as follows. For any i such that
0 < i < n, if the layer Li is thin then set
δi = span{σi , τi },
where σi and τi are the simplices of the directed geodesics between x and y that are
contained in layer Li . For any i such that the layer Li is thick, consider the thick
interval (j, k) that contains i and put
δi = S(ρi ),
where ρi is the simplex of Euclidean diagonal that is contained in the layer i in the
characteristic disk for (j, k), and S denotes the characteristic mapping. Finally let
δ0 = x and δn = y.
By definition, consecutive simplices of the Euclidean geodesic (δi )ni=0 are contained
in consecutive layers between x and y. Unlike for directed geodesics, not every two
consecutive simplices δi , δi+1 span a simplex of X. However, the following holds.
Proposition 5.3. [OP09, Remark 3.1] Suppose (δi )ni=0 is a Euclidean geodesic between vertices x and y. Then there exists a sequence of vertices (vi )ni=0 such that
vi ∈ δi and (vi )ni=0 is a geodesic.
Also note that Euclidean geodesics are symmetric with respect to their endpoints.
We now present two theorems describing the crucial properties of Euclidean geodesics.
The first one, roughly speaking, says that Euclidean geodesics are coarsely closed
under taking subsegments. The second one is a coarse form of a CAT(0) inequality
for Euclidean geodesics.
Theorem 5.4. [OP09, Theorem B] Let (δi )ni=0 be a Euclidean geodesic between vertices x and y. Take j, k ∈ {0, . . . , n} with j < k and let (ri )ki=j be a geodesic such that
ri ∈ δi for i ∈ {j, . . . , k}. Let (δij,k )ki=j denote the Euclidean geodesic between vertices

rj and rk . Then for every i ∈ {j, . . . , k} for any vertices vi ∈ δi and ui ∈ δij,k we have
d(vi , ui ) 6 C,
where C > 0 is a universal constant.

Theorem 5.5. [OP09, Theorem C] Let x, y and ye be vertices of X with d(x, y) = n
vi )m
and d(x, ye) = m. Let (vi )ni=0 and (e
i=0 be geodesics such that for all appropriate i we
20

HYPERBOLIC ISOMETRIES AND BOUNDARIES

have vi ∈ δi and vei ∈ δei , where (δi )ni=0 and (δei )m
i=0 are the Euclidean geodesics between
x and y and between x and ye respectively. Then for any 0 6 c 6 1 we have
d(v⌊cn⌋ , ve⌊cm⌋ ) 6 c · d(vn , vem ) + C,
where C > 0 is a universal constant.
5.7. Good geodesics. From now on let C > 0 be a fixed constant which satisfies
the assertions of both Theorem 5.4 and Theorem 5.5. In particular this means that
C > 200 ([OP09, p. 2877]). Having an explicit lower bound will be needed in Section 6.
Theorem 5.4 presents a model behaviour, which motivates the definition of good
geodesics.
Definition 5.6 (C ′ –good geodesic). Let (vi )ni=0 be a geodesic in X. For j, k ∈
{0, . . . , n} let (δij,k )ki=j denote the Euclidean geodesic between vertices vj and vk . We
say that (vi )ni=0 is a C ′ –good geodesic if for every two vertices vj and vk , for every
i ∈ {j, . . . , k} for any vertex ui ∈ δij,k we have
d(vi , ui ) 6 C ′ ,
where C ′ > 0 is a positive constant. An infinite geodesic is a C ′ –good geodesic if
every of its finite subgeodesics is a C ′ –good geodesic. Observe that for a C ′ –good
geodesic any of its subgeodesics is a C ′ –good geodesic as well.
Definition 5.7 (Good geodesic). A geodesic (vi )i (finite or infinite) is a good geodesic
if it is a C–good geodesic.
In particular, by Theorem 5.4 any geodesic arising from Proposition 5.3 is a good
geodesic. Consequently, any two vertices of X can be joined by a good geodesic (cf.
[OP09, Corollary 3.3]).
We finish this section with the following two remarks.
Remark 5.8. By going through the steps of the construction, one observes that the
directed, Euclidean, and good geodesics are preserved by simplicial automorphisms
of X.
Remark 5.9. The main goal of the construction outlined in this section is to establish
Theorem 2.8. This theorem plays the key role in showing various properties of the
boundary in [OP09]. Theorem 2.8 follows easily from Definition 5.7, Proposition 5.3
and Theorem 5.5. In particular, the constant D appearing in Theorem 2.8 may be
taken to be 3C.
21

TOMASZ PRYTULA

6. Fixed points on the boundary
The purpose of this section is to show that every hyperbolic isometry h of a systolic
complex X fixes a pair of points on the boundary. These two points, denoted by h+∞
and h−∞ are the canonical fixed points of h, in the sense that for any vertex x ∈ X
we have (hn · x)n → h+∞ and (h−n · x)n → h−∞ as n → ∞ in X = X ∪ ∂X.
To obtain h+∞ and h−∞ we show that there exists a bi-infinite good geodesic γ
such that γ and h · γ are asymptotic. In principal, one could expect a stronger result,
namely the existence of an h–invariant good geodesic. However, there are examples
of systolic complexes where a hyperbolic isometry has no invariant geodesics at all
[Els09b, Example 1.2]. It is true though, that for every hyperbolic isometry h there
is a geodesic γ, such that h · γ and γ are Hausdorff 1–close [Els09b, Theorem 1.3].
Unfortunately, in our construction the distance between h · γ and γ depends on L(h).
Theorem 6.1. Let X be a systolic complex on which a group G acts geometrically.
Let h ∈ G be a hyperbolic isometry. Then either there exists a bi-infinite good geodesic
which is contained in DispK (h) for some K = K(h) and DispK (h) is h–cocompact,
or there exists an h–invariant good geodesic.
We now state and prove the main result of this section assuming Theorem 6.1.
Proposition 6.2. Let G be a group acting geometrically on a systolic complex X and
let h ∈ G be a hyperbolic isometry. Then:

(1) there exist points h−∞ and h+∞ in the boundary ∂X which are fixed by h,
(2) for any vertex x ∈ X we have (hn · x)n → h+∞ and (h−n · x)n → h−∞ as
n → ∞ in the compactification X = X ∪ ∂O X, where O ∈ X is some base
vertex.

Since for any two vertices O, O ′ ∈ X there is a homeomorphism between X ∪ ∂O X
and X ∪ ∂O′ X (see Subsection 2.2), the choice of O does not really matter. In order
to simplify the argument we will choose O during the proof.
Proof. To show that a sequence (xn )∞
n=0 converges to a point [ξ] ⊂ ∂O X in X, it is
∞
enough to find a sequence (vn )n=0 ⊂ ξ, such that d(vn , O) → ∞ as n → ∞ and such
that d(vn , xn ) is uniformly bounded (see [OP09, Definition 4.1]).
By Theorem 6.1 there either exists a bi-infinite good geodesic γ ⊂ DispK (h) and
DispK (h) is h–cocompact, or there exists an h–invariant good geodesic γ. We first
give the proof assuming that there exists a good geodesic γ which is h–invariant.
Choose a vertex O ∈ γ and parametrise vertices of γ by integers such that γ(0) = O
and h·γ(0) = γ(L(h)). Then γ splits into two good geodesic rays γ [0,+∞] and γ [0,−∞]
22

HYPERBOLIC ISOMETRIES AND BOUNDARIES

starting at O. Define h+∞ = [γ
for every i > 0 we have
d(γ

[0,+∞]

[0,±∞]

] and h−∞ = [γ

(i), h · γ

[0,±∞]

[0,−∞]

]. Since γ is h–invariant,

(i)) = L(h),

and thus both h+∞ and h−∞ are fixed by h.
Let x ∈ X be an arbitrary vertex and let M = d(x, O). For any n > 0 we have:
(1) hn · O ∈ γ [0,+∞] ,
(2) d(hn · O, O) → ∞ as n → ∞,
(3) d(hn · x, hn · O) = M .

This implies that hn · x → [γ

[0,+∞]

] = h+∞ and h−n · x → [γ

[0,−∞]

] = h−∞ .

In the case where γ ⊂ DispK (h) is a bi-infinite good geodesic and DispK (h) is
h–cocompact we proceed similarly as above. Choosing a vertex O ∈ γ splits γ into
two good geodesic rays. We denote them by γ [0,+∞] and γ [0,−∞] , even though we did
not specify how we choose an orientation of γ (it will become clear from the proof).
Consequently, let h+∞ = [γ [0,+∞] ] and h−∞ = [γ [0,−∞] ]. Both γ [0,+∞] and γ [0,−∞]
are contained in DispK (h) and therefore for every i > 0 we have
d(γ

[0,±∞]

(i), h · γ

[0,±∞]

(i)) 6 K,

and hence h fixes both h+∞ and h−∞ .
Let x ∈ X be an arbitrary vertex and let M = d(x, O). Since DispK (h) is h–
cocompact and γ is bi-infinite it follows that there exists R > 0 such that
DispK (h) ⊂ BR (γ, X).
Consider the sequence (hn · O)n . We do not necessarily have (hn · O)n ⊂ γ

[0,∞]

, but

d(hn

for any n > 0 there exists a vertex vn ∈ γ [0,∞] with
· O, vn ) 6 R. Then, by
the triangle inequality we have d(O, vn ) → ∞ as n → ∞ since d(O, hn · O) → ∞ as
n → ∞. Finally, for any n > 0 we have
d(hn · x, vn ) 6 d(hn · x, hn · O) + d(hn · O, vn ) 6 M + R,
and therefore hn · x → [γ

[0,+∞]

] = h+∞ and h−n · x → [γ

[0,−∞]

] = h−∞ .



It remains to prove Theorem 6.1. Since the proof is fairly long we outline it first.
Outline of the proof of Theorem 6.1. Observe that the action of h on X preserves
Min(h). We consider two cases: when Min(h) is h–cocompact and when it is not
h–cocompact. These two cases will lead to the two respective claims of the theorem.
In the first case we show in Lemma 6.3 that for any two vertices x, y ∈ Min(h) the
Euclidean geodesic between these vertices is contained in DispK (h) for some K > 0.
This is achieved by showing that both directed geodesics between these vertices, and
23

TOMASZ PRYTULA

characteristic disks spanned by those geodesics, belong to DispK (h). The main tool in
the proof of Lemma 6.3 is the Fellow Traveller Property of directed geodesics. Then
we construct the desired good geodesic, roughly, as a limit of good geodesics between
vertices h−n · x and hn · x for a fixed vertex x ∈ Min(h).
In the second case, since CG (h) acts geometrically on Min(h), we deduce that
there is a hyperbolic isometry g that commutes with h, such that hg, hi ∼
= Z2 . Thus
by the systolic Flat Torus Theorem there is a flat F ⊂ X on which the subgroup
hg, hi acts by translations. Then using Lemma 6.4 we find an h–invariant C ′ –good
geodesic inside F (treated as a systolic complex on its own), for a certain C ′ > 0.
By Lemma 6.5 any C ′ –good geodesic in F is a (C ′ + 10)–good geodesic in X. In the
above procedure we are able to choose C ′ so that C ′ + 10 is less than C and therefore
the constructed (C ′ + 10)–good geodesic is a good geodesic in X.
Before giving the proof of Theorem 6.1 we state and prove the three lemmas mentioned above.
Lemma 6.3. Consider two vertices x, y ∈ Min(h) ⊂ X and let (δi )ni=0 be the
Euclidean geodesic between x and y. Then we have (δi )ni=0 ⊂ DispK (h), where
K = 9 · L(h) + 6.
Proof. Let (σi )ni=0 be a directed geodesic from x to y (i.e., σ0 = x and σn = y).
Then, by the Fellow Traveller Property [JS06, Proposition 11.2] applied to directed
geodesics (σi )ni=0 and (h · σi )ni=0 , for each i ∈ {0, . . . , n} for any vertex s ∈ σi we have
d(s, h · s) 6 3 · max{d(x, h · x), d(y, h · y)} + 1 = 3 · L(h) + 1,
since x, y ∈ Min(h). Put K ′ = 3 · L(h) + 1. By the above inequality we get that
(σi )ni=0 ⊂ DispK ′ (h).
Now let (σi )ni=0 be as above, and let (τi )ni=0 be the directed geodesic from y to x (see
Convention 5.1). Clearly, by the argument above, we also have (τi )ni=0 ⊂ DispK ′ (h).
If for some i ∈ {0, . . . , n} the layer Li is thin, then by definition δi = span{σi , τi } and
therefore it is contained in DispK ′ (h), since both σi and τi are so. If the layer Li is
thick then we proceed as follows.
Take any vertex z ∈ δi . We claim that there exist vertices si ∈ σi and ti ∈ τi and a
geodesic α between si and ti , such that z lies on α. Indeed, consider a thick interval
that contains i, let ∆ be an appropriate characteristic disk and let vi wi be a geodesic
in ∆ that is the layer i in ∆. Any characteristic surface S : ∆ → X restricted to
vi wi is an isometric embedding. Moreover, any vertex of δi lies in the image S(vi wi )
for some such surface. Take a surface S : ∆ → X such that z ∈ S(vi wi ) and put
si = S(vi ), ti = S(wi ) and let α = S(vi wi ). This proves the claim.
24

HYPERBOLIC ISOMETRIES AND BOUNDARIES

Put m = d(si , ti ) and let (ρj )m
j=0 be a directed geodesic from si to ti (actually, here
the direction is not important). Let (uj )m
j=0 be a geodesic such that uj ∈ ρj for every
j ∈ {0, . . . , m} (in particular u0 = si and um = ti ).
Since the layer Li is convex, both α and (uj )m
u=0 are contained in Li . Since Li is ∞–
large, any two geodesics with the same endpoints are Hausdorff 1–close [JŚ06, Lemma
2.3]. Therefore α and (uj )m
j=0 are 1–close. Finally, by the Fellow Traveller Property
m
(applied to (ρj )j=0 and (h · ρj )m
j=0 ) for any j ∈ {0, . . . , m} we have
d(uj , h · uj ) 6 3 · max{d(si , h · si ), d(ti , h · ti )} + 1 6 3 · K ′ + 1,
since both si and ti belong to DispK ′ (h). Because α and (uj )m
j=0 are 1–close and z ∈ α,
′
the above inequality implies that d(z, h·z) 6 3·K +1+2 and hence z ∈ Disp3K ′ +3 (h).
Since z ∈ δi was arbitrary we obtain that δi ⊂ Disp3K ′ +3 (h) for any i such that Li is
thick. This, together with the assertion that δi ⊂ DispK ′ (h) for any i such that Li is
thin, finishes the proof of the lemma, as 3 · K ′ + 3 = 9 · L(h) + 6.

In the next lemma we study the equilaterally triangulated Euclidean plane E2∆ .
We view it simultaneously as a systolic simplicial complex and as a CAT(0) metric
space (cf. Subsections 5.4 and 5.5). We denote the CAT(0) distance between points
of E2∆ by d E2 in order to distinguish it from the standard (combinatorial) distance d.
Lemma 6.4. Let h be a hyperbolic isometry of E2∆ and let γ ⊂ E2∆ be an h–invariant
geodesic. Suppose that β is a CAT(0) geodesic (i.e., a straight line) such that γ and
β are Hausdorff K–close with respect to the CAT(0) distance, for some K > 0. Then
√ + 1)–good geodesic.
γ is a ( 4K
3
Proof. We first give the idea of the proof. We observe that for any thick interval
in E2∆ there is a unique characteristic surface. After identifying the characteristic
disk ∆ with its image, the CAT(0) geodesic α in the modified characteristic disk ∆′
is uniformly close to β, and this distance depends only on K. This will imply the
lemma as the simplices of the Euclidean geodesic are 1–close to α, and γ is K–close
to β. The case of thin layers will follow easily from the methods used to prove the
case of thick intervals.
We now begin the proof. Let x and y be any two vertices of γ. Let (σi )ni=0 be the
directed geodesic from x to y and let (τi )ni=0 be the directed geodesic going in the
opposite direction (again, simplices of (τi )ni=0 are indexed in the same direction as for
(σi )ni=0 , see Convention 5.1). One checks that these geodesics have the form shown
in Figure 1.
For k ∈ {0, 1, . . . , n} let lk denote the infinite line in E2∆ that contains σk and τk .
In particular, for k ∈ {1, . . . , n − 1} the line lk contains the layer Lk between x and y.
25

TOMASZ PRYTULA

σ2m σ2m+1

σn−2 σn−1

y

(σi )ni=0
σ2m−1

σ1
x

σ3
σ2
τ1 τ2

τn−2 τn−1

(τi )ni=0
τn−2m+2
τn−2m+1

τn−2m

l2m+1

Figure 1. Generic form and position of the directed geodesics (σi )ni=0
and (τi )ni=0 in E2∆ .

Note that the geodesic (σi )ni=0 splits into two parts: the part (σi )2m
i=0 , where σi is a
n
vertex for even i and an edge for odd i, and the part (σi )i=2m+1 , which consists entirely
of vertices. Now observe that if 2m ∈ {0, 2, n − 1, n} then for every k ∈ {1, . . . , n − 1}
the layer Lk with respect to (σi )ni=0 and (τi )ni=0 is thin. If 2m = n − 2 then for k ∈
{1, 2m+1} the layer Lk is thin, and for k ∈ {2, . . . , 2m} the layer Lk is thin for k even,
and thick of thickness 2 for k odd. The above five cases (2m ∈ {0, 2, n − 2, n − 1, n})
will be dealt with at the end.
Now assume that 2 < 2m < n − 2. In this case for k ∈ {1, 2, n − 1, n − 2} layers
Lk are thin and (2, n − 2) is a thick interval (i.e., layers L3 , L4 , . . . , Ln−3 are thick).
Let S : ∆ → E2∆ be a characteristic surface for the interval (2, n − 2). The image of S
is presented in Figure 2. Observe that S is the unique characteristic surface for this
interval, and that it is an isometric embedding. Therefore we can identify ∆ with
S(∆) and treat it as a subcomplex of E2∆ .
Let α denote the CAT(0) geodesic in the modified characteristic disc ∆′ ⊂ ∆ ⊂ E2∆
between the midpoints of edges [σ2 , τ2 ] and [σn−2 , τn−2 ]. Denote these midpoints by
v2 and vn−2 respectively. Let α′ be a CAT(0) geodesic joining x and y. Both α and
α′ are shown in Figure 2. Note that α′ does not appear in the construction of the
Euclidean diagonal; it is an auxiliary geodesic that will be used to estimate distances
between α and β.
Introduce a coordinate system on E2∆ by setting x as the base vertex, and vectors
−
→ and −
→ where w is the vertex of σ that does not belong to τ , as the base vectors
xτ
xw,
1
1
1
(see Figure 2). Note that both α and α′ are contained in the sector of E2∆ bounded
→ and −
→ Moreover, the
by x and half-lines emanating from x in the directions of −
xτ
xw.
1
→ and the Euclidean angle between α′ and −
→ are
Euclidean angle between α and −
xτ
xτ
1
1
26

HYPERBOLIC ISOMETRIES AND BOUNDARIES

S(∆)
∆′

(σi )ni=0

σ3
σ2
α′
σ1
α
q2
−
→
xw
v
2
x
−
→τ1 τ2
xτ
1

σ2m σk(= σ2m+1 ) σn−2 σn−1 y
vn−2
qn−2
σ2m−1 v
τn−2 τn−1
k
qk
(τi )ni=0
τn−2m+2
τn−2m+1

τn−2m

lk

Figure 2. Image of the characteristic surface S : ∆ → E2∆ and the
modified characteristic disc ∆′ . Geodesics α and α′ and the coordinate
system. The CAT(0) distance between vk and qk is at most 12 .
both between 0◦ and 30◦ . In particular α′ intersects the interior of edges [σ2 , τ2 ] and
[σn−2 , τn−2 ]. Call the intersection points q2 and qn−2 respectively.
Note that the CAT(0) distances between v2 and q2 and between vn−2 and qn−2 are
less than 12 . For k ∈ {2, . . . , n − 2} let qk and vk denote the intersection points of
α and α′ respectively with the layer Lk (see Figure 2). Since the edges [σ2 , τ2 ] and
[σn−2 , τn−2 ] are parallel to all the layers, by elementary Euclidean geometry, for any
k ∈ {2, . . . , n − 2} we have:
d E2 (qk , vk ) 6 max{d E2 (q2 , v2 ), d E2 (qn−2 , vn−2 )} 6

1
.
2

(6.1)

Now consider the CAT(0) geodesic β. First we determine the position of β with
→ and −
→ Since both −
→ and −
→ coordinates of y are
respect to the base vectors −
xτ
xw.
xw
xτ
1
1
greater than the coordinates of x, and since γ is h–invariant, it follows that there
are vertices z ∈ γ such that (x, y, z) lie on γ in this order and both coordinates of z
are arbitrarily large. Since β stays K–close to γ, it follows that the Euclidean angle
→ to β is between 0◦ and 60◦ . Examples of possible
directed counter-clockwise from −
xτ
1
geodesics β and γ are shown in Figure 3.
It follows that β intersects every line lk at a single point, which we denote by pk .
For any k ∈ {1, . . . , n − 1} let tk be a vertex of γ that belongs to layer Lk (and hence
to the line lk ), and let t0 = x and tn = y. By assumption, geodesic β passes within
→ to β is
the CAT(0) distance K from any tk . Then, since the directed angle from −
xτ
1
between 0◦ and 60◦ , we claim that for any k ∈ {0, . . . , n} we have:
2K
d E2 (pk , tk ) 6 √ .
3
27

(6.2)

TOMASZ PRYTULA

(σi )ni=0

σ3
σ2
α′
σ1
α
q2
−
→
xw
v
2
x
−
→τ1 τ2
xτ
1
β
γ
p0
l0

σ2m σk(= σ2m+1 ) σn−2 σn−1 y
vn−2
qn−2
pn
τn−2 τn−1
σ2m−1 v
k
qk
pk
ln
τn−2m+2

tk

(τi )ni=0

τn−2m+1
τn−2m
lk

Figure 3. Examples of (parts of) geodesics β and γ and their position
with respect to α and α′ . Distances between vertices vk , qk , pk , tk are
measured on the line lk .
K
√ (=
To obtain the constant 2K
cos30◦ ) one checks the extreme cases where β is parallel
3
→ or −
→ In particular we have d 2 (p , x) 6 2K
√ and d 2 (p , y) 6 2K
√ .
to either −
xτ
xw.
1

0

E

3

E

n

3

Now since α′ connects x and y and since β passes through p0 and pn it follows that
for k ∈ {0, . . . , n} we have:
2K
d E2 (pk , qk ) 6 √ .
(6.3)
3
Denote by (δi )ni=0 the Euclidean geodesic between x and y. By definition, for thin
layers we have δ1 = σ1 , δ2 = [σ2 , τ2 ], δn−2 = [σn−2 , τn−2 ] and δn−1 = τn−1 . For thick
layers we have δi = ui where ui ∈ Li is a vertex that is at distance less than 12 from
vi or δi = [ui , u′i ] if ui , u′i ∈ Li and d E2 (ui , vi ) = d E2 (u′i , qk ) = 21 . In any case, for
k ∈ {3, . . . , n − 3} for any vertex uk ∈ δk we have:
d E2 (uk , vk ) 6

1
.
2

(6.4)

Finally, by combining (6.1), (6.2), (6.3) and (6.4) for any k ∈ {3, . . . , n − 3} for any
uk ∈ δk we have:
d E2 (tk , uk ) 6 d E2 (tk , pk ) + d E2 (pk , qk ) + d E2 (qk , vk ) + d E2 (vk , uk )
2K
2K
1 1
6 √ +√ + +
3
3 2 2
4K
= √ + 1.
3

(6.5)

Since all the distances above are measured on the line lk , the same estimate holds for
the standard (combinatorial) distance.
28

HYPERBOLIC ISOMETRIES AND BOUNDARIES

For k ∈ {2, n − 2} we have δk = [σk , τk ] and one observes that d E2 (qk , σk ) 6 1 and
d E2 (qk , τk ) 6 1. Combining this with (6.2) and (6.3) we obtain the same estimate as
in (6.5). Now for k ∈ {1, n − 1}, let qk be the point of intersection α′ ∩ δk . By a
direct observation we see that for any vertex uk ∈ δk we have dE2 (vk , uk ) 6 1, and
thus again we obtain the same estimate as in (6.5).
We conclude that for any k ∈ {1, . . . , n − 1} for any uk ∈ δk we have
4K
d(tk , uk ) 6 √ + 1.
3

√ + 1)–good geodesic.
Since x and y were arbitrary, this proves that γ is a ( 4K
3

For the remaining cases, where geodesics (σi )ni=0 and (τi )ni=0 are close to each other
(2m ∈ {0, 2, n − 2, n − 1, n}), we proceed analogously. One observes that the auxiliary
CAT(0) geodesic α′ joining x and y passes at the CAT(0) distance at most 1 from
all the vertices of (δi )ni=0 in all the appropriate layers. The rest of the argument goes
the same as in the first case (i.e., one combines the above observation with (6.2) and
(6.3) and obtains the same estimate as in (6.5)).

A flat in a systolic complex X is an isometric embedding F : E2∆ ֒→ X.
Lemma 6.5. Let F : E2∆ ֒→ X be a flat, and suppose that γ ⊂ E2∆ is a C ′ –good
geodesic (where E2∆ is treated as a systolic complex on its own). Then F (γ) is a
(C ′ + 10)–good geodesic in X.
2
Proof. Pick any two vertices x, y ∈ E2∆ , let (δi )m
i=0 ⊂ E∆ be the Euclidean geodesic
between x and y in E2∆ , and let (δei )m
i=0 ⊂ X be the Euclidean geodesic between F (x)
and F (y) in X. To prove the lemma it is enough to prove the following claim.

Claim. For any i ∈ {0, 1, . . . , m} for any two vertices zi ∈ δi and zei ∈ δei we have
d(F (zi ), zei ) 6 10.

The rest of the argument is devoted to proving the claim. Let (σi )ni=0 and (τi )ni=0
be directed geodesics in E2∆ going respectively from x to y and from y to x, and let
(e
σi )ni=0 and (e
τi )ni=0 be the corresponding directed geodesics between F (x) and F (y)
in X (see Convention 5.1). As in Lemma 6.4, we will first deal with the case when
there is a single thick interval for (σi )ni=0 and (τi )ni=0 , i.e., with the notation from
Lemma 6.4, for (σi )ni=0 we assume that 2 < 2m < n − 2.
Consider the thick interval (2, n − 2) and for every i ∈ {2, 3, . . . , n − 2} choose
vertices si ∈ σi and ti ∈ τi that realise the thickness of layer i in E2∆ . Let S : ∆ → E2∆
be a characteristic surface for the cycle
α = (s2 , s3 , . . . , sn−2 , tn−2 , tn−1 , . . . , t2 , s2 ).
29

TOMASZ PRYTULA

Observe that S is the unique characteristic surface for the interval (2, n − 2), and
that it is an isometric embedding. Therefore we will identify the characteristic disk
∆ with its image S(∆) ⊂ E2∆ .
The map F ∆ : ∆ → X does not have to be a characteristic surface for geodesics
(e
σi )ni=0 and (e
τi )ni=0 , e.g., not all of the vertices of F (α) belong to the appropriate
simplices of (σei )ni=0 and (e
τi )ni=0 . A priori we do not even know whether (2, n − 2) is a
thick interval for these geodesics.
We will now show how to modify F to obtain a characteristic surface for (e
σi )ni=0
and (e
τi )ni=0 . The idea is that the images F ((σi )ni=0 ) and F ((τi )ni=0 )) are 1–close to
the geodesics (σei )ni=0 and (e
τi )ni=0 respectively, and therefore a small perturbation of
the map F would give the desired characteristic surface. We will now make this
idea precise. For every i ∈ {2, 4, 6, . . . , 2m} choose any vertex ui ∈ σ
ei and for every
i ∈ {n − 2m, n − 2m + 2, n − 2m + 4, . . . , n − 2} choose any vertex wi ∈ τei . Then by
[Els09c, Lemma 3.9] (applied simultaneously to geodesics (σi )ni=0 and (τi )ni=0 ) there
exists a flat F ′ : E2∆ → X such that:
(1)
(2)
(3)
(4)

for
for
for
for

i ∈ {2, 4, . . . , 2m} we have F ′ (σi ) = ui ,
i ∈ {n − 2m, n − 2m + 2, . . . , n − 2} we have F ′ (τi ) = wi ,
every vertex x ∈ E2∆ not considered in (1) and (2), we have F ′ (x) = F (x),
every 0 6 i 6 n we have F ′ (σi ) = Im(F ′ ) ∩ σ
ei and F ′ (τi ) = Im(F ′ ) ∩ τei .

Images of F and F ′ are shown in Figure 4.
u2m

F (y)

u2m−2
F (E2∆ )

u2i

u4
u2
F (x)

F (σ2 )

wn−2

F (σ2m )

wn−4

F (σ2m−2 )
w2i
F (σ2i )
wn−2m+2

F (σ4 )
wn−2m

F (τn−2 )
F (τn−4 )
F (τ2i )

F (τn−2m+2 )
F (τn−2m )

Figure 4. Images of flats F and F ′ . The part of F ′ (E2∆ ) which differs
from F (E2∆ ) is red.
Observe that for i ∈ {2, 4, . . . , 2m} vertices F (σi ) and F ′ (σi ) = ui are connected.
Indeed, since F and F ′ agree on all the neighbours of σi in E2∆ , taking a pair of
neighbours v1 , v2 ∈ E2∆ of σi that are not adjacent gives a 4–cycle
(F (v1 ), F (σi ), F (v2 ), F ′ (σi )).
30

HYPERBOLIC ISOMETRIES AND BOUNDARIES

Since X is 6–large, this cycle has a diagonal. It cannot be [F (v1 ), F (v2 )] since F is an
isometric embedding, and v1 and v2 are not adjacent. Thus it must be [F (σi ), F ′ (σi )].
Analogously, for any i ∈ {n − 2m, n − 2m + 2, . . . , n − 2} vertices F (τi ) and F ′ (τi ) =
wi are connected. The above assertions, together with property (3) of the map F ′ ,
imply that for any vertex x ∈ E2∆ we have
d(F (x), F ′ (x)) 6 1.
We claim that for any 2 6 i 6 n − 2 the vertices F ′ (si ) ∈ σ
ei and F ′ (ti ) ∈ τei realise
the thickness of layer i for (e
σi )ni=0 and (e
τi )ni=0 in X, and so layers L3 , L4 , . . . , Ln−3 are
thick, and layers L2 and Ln−2 are thin. This follows essentially from the fact that
F ′ : E2∆ → X is an isometric embedding. Similarly, one can show that layers L1 and
Ln−1 in X are thin. Finally, we conclude that
F′

∆

:∆→X

is a characteristic surface for the thick interval (2, n − 2) for geodesics (e
σi )ni=0 and
n−3
(e
τi )ni=0 in X. Let (ρi )i=3
be the Euclidean diagonal for ∆ ⊂ E2∆ and denote by (δi )m
i=0
2
the Euclidean geodesic between vertices x and y in E∆ . We have δi = span{σi , τi } for
i ∈ {1, 2, n−2, n−1}. For all remaining i we have δi = ρi (since ∆ ∼
= S(∆) ⊂ E2∆ is the
unique characteristic surface for the thick interval (2, n−2)). For any i ∈ {0, 1, . . . , n},
for any vertex zi ∈ δi we have
d(F (zi ), F ′ (zi )) 6 1.
For any i ∈ {0, 1, . . . , n}, for any vertices zi ∈ δi and zei ∈ δei we claim that
d(F ′ (zi ), zei ) 6 1.

(6.6)

(6.7)

This follows for i ∈ {1, 2, n−2, n−1} from the property (4) of the map F ′ : E2∆ → X.
Namely we have that F ′ (σi ) ⊂ σ
ei and F ′ (τi ) ⊂ τei , and by definition δi = span{σi , τi }
and δei = span{e
σi , τei }. For i ∈ {3, 4, . . . , n − 3} by definition of a Euclidean geodesic
and the fact that F ′ ∆ is a characteristic surface for (e
σi )ni=0 and (e
τi )ni=0 we obtain
that F ′ (ρi ) ⊂ δei .
Finally, combining (6.6) and (6.7), for any i ∈ {0, 1, . . . , n} for any two vertices
zi ∈ δi and zei ∈ δei we have
d(F (zi ), zei ) 6 2.

This finishes the proof of the claim under the assumption that 2 < 2m < n − 2.
Now assume 2m ∈ {0, 2, n − 2, n − 1, n}. In this case any layer Li ⊂ E2∆ has
thickness at most 2. By [Els09c, Proposition 3.8] (which is a weaker formulation
of [Els09c, Lemma 3.9] used above) for any vertices si ∈ σi and ui ∈ σ
ei we have
d(F (si ), ui ) 6 2. The same estimate holds for vertices of τi and τei . It follows from
31

TOMASZ PRYTULA

the triangle inequality, that for any i ∈ {0, 1, . . . , n} the thickness of the layer i in X
is at most 6.
Observe that by definition of the Euclidean geodesic, any simplex δi lies between
simplices σi and τi in the layer Li ⊂ E2∆ . More precisely, the distance between any
vertex zi ∈ δi and any vertex ui ∈ σi is less than the thickness of Li . Clearly the same
estimate holds for vertices of δei and σ
ei , if one replaces thickness of Li by thickness
of layer i in X. From these considerations we conclude that for any i ∈ {0, 1, . . . , n},
for any zi ∈ δi and zei ∈ δei and for any vertices si ∈ σi and ui ∈ σei we have:
d(F (zi ), zei ) 6 d(F (zi ), F (si )) + d(F (si ), ui ) + d(ui , zei ) 6 2 + 2 + 6 = 10.
This estimate is by no means optimal.



We are ready now to prove Theorem 6.1.
Proof of Theorem 6.1. Case 1: Min(h) is h–cocompact. Let K be the constant
appearing in Lemma 6.3. Since Min(h) is h–cocompact, by Lemma 4.2 the subcomplex DispK (h) is h–cocompact as well. Pick a vertex x ∈ Min(h) ⊂ DispK (h), and for
any n > 0 consider vertices h−n · x, hn · x ∈ Min(h). Note that d(h−n · x, hn · x) is not
necessarily equal to 2n · L(h), but we can assume that it is even (by passing to a subsequence of the form ni = ik for some k > 1 if necessary, see [Els09b, Theorem 1.1]).
n
Put mn = 12 · d(h−n · x, hn · x) and let (δin )m
i=−mn be the Euclidean geodesic between
n
h−n · x and hn · x. By Lemma 6.3 we have (δin )m
i=−mn ⊂ DispK (h). Since DispK (h) is
n
h–cocompact, there exists R > 0 such that for every n the geodesic (δin )m
i=−mn intern
sects the ball BR (x, X). Let in be an integer such that δinn is a simplex of (δin )m
i=−mn
that intersects BR (x, X) (such in is not unique in general, we choose one for each n).
By replacing R with R + 1 we can assume that δinn ⊂ BR (x, X).
Since the ball BR (x, X) contains only finitely many simplices, there are infinitely
many n such that δinn is equal to a fixed simplex of BR (x, X). Denote this simplex
n
by δe0 . Now since the sphere S1 (δe0 , X) is finite, among geodesics (δin )m
i=−mn for which
δinn = δe0 there are infinitely many such that δinn +1 is equal to a fixed simplex δe1
and δinn −1 is equal to a fixed simplex δe−1 . By continuing this procedure for spheres
Sk (δe0 , X) for k > 1, we obtain a bi-infinite sequence of simplices
(δei )∞
i=−∞ ⊂ DispK (h),
such that any of its finite subsequences is a Euclidean geodesic.
By Proposition 5.3 for any finite subsequence, say (δei )m
i=−m , there exists a geodesic
m
γm = (vi )i=−m such that vi ∈ δei . By Theorem 5.4 any γm is a good geodesic. By a
diagonal argument, from the sequence (γm )∞
m=0 we can extract a bi-infinite geodesic
γ = (vi )∞
i=−∞ , which is a good geodesic, as any of its finite subgeodesics is contained in
32

HYPERBOLIC ISOMETRIES AND BOUNDARIES

a good geodesic γm for some m > 0. Since for every i ∈ N we have vi ∈ δei ⊂ DispK (h),
we conclude that γ ⊂ DispK (h).
Case 2. Min(h) is not h–cocompact. By [OP16, Corollary 5.8] the centraliser
CG (h) is commensurable with the product Fn ×Z, such that the subgroup hhi ⊂ CG (h)
is commensurable with the ‘Z’ factor of the latter. By Theorem 4.3 the group CG (h)
acts cocompactly on Min(h). Since Min(h) is not h–cocompact, we conclude that
n > 1, and so there exists an element g ∈ CG (h) such that hg, hi ∼
= Z2 . By the Flat
Torus Theorem ([Els09, Theorem 6.1]) there exists a flat F : E2∆ → X whose image
is preserved by the action of hg, hi. We will now construct an h–invariant geodesic
γ ⊂ F (E2∆ ) which satisfies the assumptions of Lemma 6.4.
Take any vertex x ∈ F (E2∆ ) and consider a CAT(0) geodesic γ ′ in F (E2∆ ) that passes
through vertices x and h · x. The isometry h acts on F (E2∆ ) ∼
= E2 as a translation
along γ ′ by distance equal to the CAT(0) length of segment γ ′ [x,h·x]. Let α be any
(combinatorial) geodesic between x and h · x that is Hausdorff 1–close to γ ′ [x,h·x].
(To obtain such α one proceeds similarly as when defining the Euclidean diagonal in
a characteristic disk in Subsection 5.5.) Define γ as
[
(hn · α).
γ=
n∈Z

By definition γ is an h–invariant geodesic, that is Hausdorff 1–close to a CAT(0)
geodesic γ ′ in F (E2∆ ). By Lemma 6.4 we get that γ is a ( √43 + 1)–good geodesic in
F (E2∆ ). Lemma 6.5 implies that γ is a ( √43 + 11)–good geodesic in X. This implies

that γ is a good geodesic in X since we have √43 + 11 < C, where C is the constant
appearing in Definition 5.7 (cf. the discussion at the beginning of Subsection 5.7). 
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E-mail address: t.p.prytula@soton.ac.uk

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