BOUNDARY CONVEX COCOMPACTNESS AND STABILITY OF
SUBGROUPS OF FINITELY GENERATED GROUPS

arXiv:1607.08899v1 [] 29 Jul 2016

MATTHEW CORDES AND MATTHEW GENTRY DURHAM

Abstract. A Kleinian group Γ < Isom(H3 ) is called convex cocompact if any orbit of Γ in H3 is
quasiconvex or, equivalently, Γ acts cocompactly on the convex hull of its limit set in ∂H3 .
Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition
above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups.
Using the Morse boundary, we develop an equivalent characterization of subgroup stability which
generalizes the above boundary characterization from Kleinian groups.

1. Introduction
There has been much recent interest in generalizing salient features of Gromov hyperbolic spaces
to more general contexts, including their boundaries and convexity properties of nicely embedded
subspaces. Among these are the Morse property, its generalization to subspaces, stability, and the
Morse boundary. In this article, we use the Morse boundary to prove that stability for subgroups of
finitely generated groups is naturally a convex cocompactness condition in the classical boundary
sense. We begin with some motivation from Kleinian groups and mapping class groups.
A nonelementary discrete (Kleinian) subgroup Γ < PSL2 (C) determines a minimal Γ-invariant
closed subspace Λ(Γ) of the Riemann sphere called its limit set and taking the convex hull of Λ(Γ)
determines a convex subspace of H3 with a Γ-action. A Kleinian group Γ is called convex cocompact
if it acts cocompactly on this convex hull or, equivalently, any Γ-orbit in H3 is quasiconvex. Another
equivalent characterization is that such a Γ has a compact Kleinian manifold; see [Mar74, Sul85].
Originally defined by Farb-Mosher [FM02] and later developed further by Kent-Leininger [KL08]
and Hamenstädt [Ham05], a subgroup H < Mod(S) is called convex cocompact if and only if any
H-orbit in T (S), the Teichmüller space of S with the Teichmüller metric, is quasiconvex, or H acts
cocompactly on the weak hull of its limit set Λ(H) ⊂ PML(S) in the Thurston compactification of
T (S). This notion is important because convex cocompact subgroups H < Mod(S) are precisely
those which determine Gromov hyperbolic surface group extensions.
In both of these examples, convex cocompactness is characterized equivalently by both a quasiconvexity condition and an asymptotic boundary condition. In [DT15b], Taylor and the second
author introduced stability in order to characterize convex cocompactness in Mod(S) by a quasiconvexity condition intrinsic to the geometry of Mod(S). In fact, stability naturally generalizes the
above quasiconvexity characterizations of convex cocompactness to any finitely generated group.
In this article, we use the Morse boundary to define an asymptotic property for subgroups
of finitely generated groups called boundary convex cocompactness which generalizes the classical
boundary characterization of convex cocompactness from Kleinian groups. Our main theorem is:
Theorem 1.1. Let G be a finitely generated group. Then H < G is boundary convex cocompact if
and only if H is stable in G.
Date: August 1, 2016.
1

Before moving on to the definitions, we discuss the situation in hyperbolic groups.
Let H be a quasiconvex subgroup of a hyperbolic group G. Then H has a limit set Λ(H) ⊂ ∂Gr G,
the Gromov boundary of G, and one can define the weak hull of Λ(H) to be the union of all geodesics
in G connecting distinct points in Λ(H). Swenson [Swe01] proved that H acts cocompactly on this
weak hull if and only H is quasiconvex in G. Hence a quasiconvex subgroup of a hyperbolic group
satisfies a boundary characterization of convex cocompactness intrinsic to the ambient geometry.
Stability generalizes quasiconvexity to any finitely generated group, and similarly the Morse
boundary generalizes the Gromov boundary. Thus Theorem 1.1 is a generalization of the hyperbolic
case to any finitely generated group.
Stability and the Morse boundary. We start with the definition of a Morse quasigeodesic:
Definition 1.2 (Morse quasigeodesic). A quasigeodesic γ in geodesic metric space X is called
N -Morse if there exists a function N : R≥1 × R≥0 → R≥0 such that if q is any (K, C)-quasigeodesic
with endpoints on γ, then q ⊂ NN (K,C) (γ), the N (K, C)-neighborhood of γ.
We call N the Morse gauge of γ.
Note that if X is hyperbolic, then every geodesic in X is Morse with a uniform gauge.
If G is a finitely generated group, then we call g ∈ G a Morse element if its orbit in any Cayley
graph of G is a Morse quasigeodesic. Some examples of Morse elements include rank-1 elements of
CAT(0) groups [BC12], pseudo-Anosov elements of mapping class groups [Beh06], fully irreducible
elements of the outer automorphism groups of free groups [AK11], and rank-one automorphisms of
hierarchically hyperbolic spaces [DHS16].
We can now give a formal definition of stability:
Definition 1.3 (Stability). If f : X → Y is a quasiisometric embedding between geodesic metric
spaces, we say X is a stable subspace of Y if there exists a Morse gauge N such that every pair of
points in X can be connected by an N -Morse quasigeodesic in Y ; we call f a stable embedding.
If H < G are finitely generated groups, we say H is stable in G if the inclusion map i : H ,→ G
is a stable embedding.
We note that stable subgroups are always hyperbolic and quasiconvex regardless of the chosen
word metric on G—stability and quasiconvexity coincide in hyperbolic spaces—and stability is
invariant under quasiisometric embeddings [DT15b].
Introduced by the first author in [Cor15] for proper geodesic spaces and later refined and generalized to geodesic spaces in [CH16], the Morse boundary of a geodesic metric space X, denoted ∂M Xe ,
consists, roughly speaking, of asymptotic classes of sequences of points which can be connected to
a fixed basepoint e ∈ X by Morse geodesic rays; see Subsection 2.2 for the formal definition. Importantly, it is a visual boundary, generalizes the Gromov boundary when X is hyperbolic, and
quasiisometries induce homeomorphisms at the level of Morse boundaries.
Boundary convex cocompactness. Let G be a finitely generated group acting by isometries on
a proper geodesic metric space X. Fix a basepoint e ∈ X. One can define a limit set Λe (G) ⊂ ∂M Xe
as the set of points which can be represented by sequences of G-orbit points; note that Λe (G) is
obviously G-invariant. One then defines the weak hull He (G) of Λe (G) in X by taking all geodesics
with distinct endpoints in Λe (G). See Section 3 for the precise definitions.
Definition 1.4 (Boundary convex cocompactness). We say that G acts boundary convex cocompactly on X if the following conditions hold:
(1) G acts properly on X;
(2) For some (any) e ∈ X, Λe (G) is nonempty and compact;
(3) For some (any) e ∈ X, the action of G on He (G) is cocompact.
2

Definition 1.5 (Boundary convex cocompactness for subgroups). Let G be a finitely generated
group. We say H < G is boundary convex cocompact if H acts boundary convex cocompactly on
any Cayley graph of G with respect to a finite generating set.
Theorem 1.1 is an immediate consequence of the following stronger statement:
Theorem 1.6. Let G be a finitely generated group acting by isometries on a proper geodesic metric
space X. Then the action of G is boundary convex cocompact if and only if some (any) orbit of G
in X is a stable embedding.
In either case, G is hyperbolic and any orbit map orbe : G → X extends continuously and Gequivariantly to an embedding of ∂Gr G which is a homeomorphism onto its image Λe (G) ⊂ ∂M Xe .
We note that Theorem 1.1 and [DT15b, Proposition 3.2] imply that boundary convex cocompactness is invariant under quasiisometric embeddings.
Remark 1.7 (On the necessity of the conditions in Definition 1.4). The compactness assumption
on Λe (G) is essential: Consider the group G = Z2 ∗ Z ∗ Z = ha, bi ∗ hci ∗ hdi acting on its Cayley
graph with the subgroup H = ha, b, ci. Since the H is isometrically embedded and convex in G, it
follows that ∂M He ∼
= Λe (H) ⊂ ∂M Ge and He (H) = H for any e ∈ G, whereas H is not hyperbolic
and thus not stable in G.
While compactness of Λe (G) does imply that He (G) is stable in X (Proposition 4.2), it is unclear
how to leverage this fact into proving properness or cocompactness of the G-action, even in the
presence of one or the other.
Remark 1.8 (Stability versus boundary convex cocompactness). We expect that stability will be
a much condition to check in practice than boundary convex cocompactness. Our purpose is to
prove that stability generalizes multiple classical notions of convex cocompactness and is thus the
correct generalization of convex cocompactness to finitely generated groups.
Remark 1.9 (Conical limit points). In addition to the above discussed notions for Kleinian groups,
there is characterization for convex cocompactness in terms of limit points, namely that every limit
point is conical. Kent-Leininger develop a similar characterization of this for subgroups of Mod(S)
acting on T (S)∪PML(S). We believe that there is a similar conicality characterization for subgroup
stability with respect to limit sets in the Morse boundary. However, we have chosen not to pursue
this here in interest of brevity. Moreover, it’s unclear how useful such a characterization would be.
Stability and boundary convex cocompactness in important examples. Since stability
is a strong property, it is interesting to characterize and produce stable subgroups of important
groups. There has been much recent work to do so, which we will now briefly overview.
(1) For a relatively hyperbolic group (G, P), Aougab, Taylor, and the second author [ADT16]
prove that if H < G is finitely generated and quasiisometrically embeds in the associated
coned space [Far98], then H is stable in G. Moreover, if we further assume that the
subgroups in P are one-ended and have linear divergence, then quasiisometrically embedding
in the coned space is equivalent to stability.
(2) For Mod(S), that subgroup stability and convex cocompactness in the sense of [FM02] are
equivalent was proven by Taylor and the second author [DT15b].
(3) For the right-angled Artin group A(Γ) of a finite simplicial graph Γ which is not a join,
Koberda-Mangahas-Taylor [KMT14] proved that stability for H < A(Γ) is equivalent to
H being finitely generated and purely loxodromic, i.e. each element acts loxodromically
on the associated extension graph Γe , a curve graph analogue. They also prove that stable
subgroups satisfy a strictly weaker condition called combinatorial quasiconvexity, sometimes
called convex cocompactness [Hag08]; we note that, unlike stable subgroups, combinatorially
quasiconvex subgroups need not be hyperbolic.
3

(4) For Out(F), the outer automorphism group of a free group on at least three letters, work
in [ADT16] proves that a subgroup H < Out(F) which has a quasiisometrically embedded
orbit in the free factor graph F is stable in Out(F). This is related to and builds on
others’ work as follows: Hamenstädt-Hensel [HH14] proved that a subgroup H < Out(F)
quasiisometrically embedding in the free factor graph is equivalent to a certain convex
cocompactness condition in the projectivization of the Culler-Vogtmann outer space and
its boundary which is analogous to that the Kent-Leininger condition on T (S) ∪ PML(S).
Following [HH14], we shall call such subgroups convex cocompact. By work of DowdallTaylor [DT14, DT15a], quasiisometrically embedding in the free factor graph implies a
stability-like property for orbit maps in outer space; we note that when such a group is also
fully atoroidal, they prove the corresponding free group extension is hyperbolic. In [ADT16],
the authors prove this stability property pulls back to genuine stability in Out(F).
We summarize these results in the following theorem:
Theorem 1.10 ([DT15b, KMT14, ADT16]). Suppose that the pair H < G satisfies one of the
following conditions:
(1) H is a quasiconvex subgroup of a hyperbolic group G;
(2) G is relatively hyperbolic and H is finitely generated and quasiisometrically embeds in the
coned space associated to G in the sense of [Far98];
(3) G = A(Γ) for Γ a finite simplicial graph which is not a join and H is finitely generated and
H quasiisometrically embeds in the extension graph Γe [KMT14];
(4) G = Mod(S) and H is a convex cocompact subgroup in the sense of [FM02];
(5) G = Out(F) and H is a convex cocompact subgroup in the sense of [HH14].
Then H is stable in G. Moreover, for (1), (3), and (4), the reverse implication also holds.
As a corollary of Theorem 1.1, we have:
Corollary 1.11. Suppose H < G are as any of (1)–(5) in Theorem 1.10. Then H is boundary
convex cocompact.
Item (1) in Corollary 1.11 is originally due to Swenson [Swe01]. In examples (2)–(5), each of
the previously established boundary characterizations of convex cocompactness was in terms of
an external space. Corollary 1.11 provides a boundary characterization of convex cocompactness
which is intrinsic to the geometry of the ambient group.
We note that Hamenstädt has announced that there are stable subgroups of Out(F) which are
not convex cocompact in the sense of [HH14], but such subgroups would be convex cocompact both
in the quasiconvex and boundary senses.
Finally, we see that boundary convex cocompactness is generic in these main example groups.
For any such group G and probability measure µ thereon, consider k ≥ 2 independent random
walks (wn1 )n∈N , . . . , (wnk )n∈N whose increments are distributed according to µ. For each n, let
Γ(n) = hwn1 , . . . , wnk i ≤ G. Following Taylor-Tiozzo [TT16], we say a random subgroup of G has a
property P if
P[Γ(n) has P ] → 1.
In the following, (1) is a consequence of [TT16], (3) of [KMT14], and (2), (4)–(5) of [ADT16]:
Theorem 1.12 ([TT16, KMT14, ADT16]). Suppose that G is any of the groups in (1)–(5) of
Theorem 1.10. Then a k-generated random subgroup of G is stable.
Hence, Theorem 1.1 gives us:
Corollary 1.13. Suppose G is any of the groups in (1)–(5) of Theorem 1.10. Then a k-generated
random subgroup of G is boundary convex cocompact.
4

Acknowledgements. We would like to thank the organizers of the “Geometry of Groups in Montevideo” conference, where part of this work was accomplished. The first and second authors were
partially supported by NSF grants DMS-1106726 and DMS-1045119, respectively. We would like
to thank Ursula Hamenstädt and Samuel Taylor for interesting conversations, and also the latter
for useful comments on an earlier draft of this paper.
2. Background
We assume the reader is familiar with basics of δ-hyperbolic spaces and their Gromov boundaries.
In this subsection we recall some of the basic definitions. For more information, see [BH99, III.H].
2.1. Gromov boundaries.
Definition 2.1. Let X be a metric space and let x, y, z ∈ X. The Gromov product of x and y with
respect to z is defined as
1
(x · y)z = (d(z, x) + d(z, y) − d(x, y)) .
2
Let (xn ) be a sequence in X. We say (xi ) converges at infinity if (xi · xj )e → ∞ as i, j → ∞. Two
convergent sequences (xn ), (ym ) are said to be equivalent if (xi · yj ) → ∞ as i, j → ∞. We denote
the equivalence class of (xn ) by lim xn .
The sequential boundary of X, denoted ∂X, is defined to be the set of convergent sequences
considered up to equivalence.
Definition 2.2. [BH99, Definition 1.20] Let X be a (not necessarily geodesic) metric space. We
say X is δ–hyperbolic if for all w, x, y, z we have
(x · y)w ≥ min {(x · z)w , (z · y)w } − δ.
If X is δ–hyperbolic, we may extend the Gromov product to ∂X in the following way:


(x · y)e = sup lim inf {(xn · ym )e } .
m,n→∞

where x, y ∈ ∂X and the supremum is taken over all sequences (xi ) and (yj ) in X such that
x = lim xi and y = lim yj .
2.2. (Metric) Morse boundary. Let X be a geodesic metric space, let e ∈ X and let N be a
(N )
Morse gauge. We define Xe to be the set of all y ∈ X such that there exists a N –Morse geodesic
[e, y] in X.
(N )

(N )

Proposition 2.3 (Xe are hyperbolic; Proposition 3.2 [CH16]). Xe
the sense of Definition 2.2.
(N )

is 8N (3, 0)–hyperbolic in
(N )

As each Xe is hyperbolic we may consider its Gromov
 boundary,
 ∂Xe , and the associated
(N )
visual metric d(N ) . We call the collection of boundaries ∂Xe , d(N ) the metric Morse boundary
of X.
(N )
Instead of focusing on sequences which live in some Xe , we now consider the set of all Morse
geodesic rays in X (with basepoint p) up to asymptotic equivalence. We call this collection the
Morse boundary of X, and denote it by ∂M X.
To topologize the boundary, first fix a Morse gauge N and consider the subset of the Morse
boundary that consists of all rays in X with Morse gauge at most N :
N
∂M
Xp = {[α] | ∃β ∈ [α] that is an N –Morse geodesic ray with β(0) = p}.

We topologize this set with the compact-open topology. This topology is equivalent to one defined
N X , which are defined as follows:
by a system of neighborhoods, {Vn (α) | n ∈ N}, at a point α in ∂M
p
5

the set Vn (α) is the set of geodesic rays γ with basepoint p and d(α(t), γ(t)) < δN for all t < n,
where δN is a constant that depends only on N .
Let M be the set of all Morse gauges. We put a partial ordering on M so that for two Morse
gauges N, N 0 ∈ M, we say N ≤ N 0 if and only if N (λ, ) ≤ N 0 (λ, ) for all λ,  ∈ N. We define the
Morse boundary of X to be
N
∂M Xp = lim ∂M
Xp
−→
M

N X is
with the induced direct limit topology, i.e., a set U is open in ∂M Xp if and only if U ∩ ∂M
p
open for all N .
In [CH16] the first author and Hume show that if X is a proper geodesic metric space, then there
(N )
NX .
is a natural homeomorphism between ∂Xe and ∂M
e

2.3. Useful facts. In this subsection, we will collect a number of basic facts and definitions.
The following lemma states that a quasigeodesic with endpoints on a Morse geodesic stays
Hausdorff close:
Lemma 2.4 (Hausdorff close; Lemma 2.1 in [Cor15]). Let X be a geodesic space and let γ : [a, b] →
X be a N -Morse geodesic segment and let σ : [a0 , b0 ] → X be a continuous (K, C)-quasi-geodesic
such that γ(a) = σ(a0 ) and γ(b) = σ(b0 ). Then the Hausdorff distance between α and β is bounded
by 2N (λ, )
The next lemma states that if a geodesic triangles ∆ has two Morse sides, then ∆ is slim and its
third side is also Morse:
Lemma 2.5 (Lemma 2.2-2.3 in [Cor15]). Let X be a geodesic space. For any Morse gauge N , there
exists a gauge N 0 such that the following holds: Let γ1 , γ2 : [0, ∞) → X be N -Morse geodesics such
that γ1 (0) = γ2 (0) = e and let x1 = γ1 (v1 ), x2 = γ2 (v2 ) be points on γ1 and γ2 respectively. Let γ be
a geodesic between x1 and x2 . Then the geodesic triangle γ1 ([0, v1 ]) ∪ γ ∪ γ2 ([0, v2 ]) is 4N (3, 0)-slim.
Moreover γ is N 0 -Morse.
The following is the standard fact that quasiisometries preserve the Gromov product:
Lemma 2.6. Suppose X and Y are proper Gromov hyperbolic metric spaces. If f : X → Y is a
quasiisometric embedding, then there exist A ≥ 1, B > 0 such that for any x, y, z ∈ X, we have
1
(x · y)z − B ≤ (f (x) · f (y))f (z) ≤ A(x · y)z + B.
A
We have the following well-known consequence:
Lemma 2.7. Suppose that f : X → Y is a quasiisometric embedding between proper Gromov
hyperbolic spaces. Then the induced map ∂f : ∂X → ∂Y is a homeomorphism onto its image.
2.4. Stability. The following is [DT15b, Definition 3.1]:
Definition 2.8 (Stability 1). if for any K ≥ 1, C ≥ 0, there exists R = R(K, C) ≥ 0 such that if
q1 : [a, b] → Y, q2 : [a0 , b0 ] → Y are (K, C)-quasigeodesics with q1 (a) = q10 (a0 ), q1 (b) = q2 (b0 ) ∈ f (X),
then
dHaus (q1 , q2 ) < R.
We will use the following definition of stability which is equivalent via Lemma 2.4:
Definition 2.9 (Stability 2). Let X, Y be geodesic metric spaces and let f : X → Y be a quasiisometric embedding. We say X is stable in Y there exists a Morse gauge N such that any x, y ∈ f (X)
are connected by a N -Morse geodesic in Y . We say that f is a stable embedding.
6

2.5. Morse preserving maps. It will be useful for having a notion of when maps between metric
spaces preserve data encoded by the Morse boundary.
Definition 2.10 (Morse preserving maps). Let X, Y be proper geodesic metric spaces and e ∈
X, e0 ∈ Y . We say that g : ∂M Xe → ∂M Ye0 Morse-preserving if for each Morse gauge N , there
(N )
(N 0 )
exists another Morse gauge N 0 such that g injectively maps ∂Xe → ∂Ye0 .
The following is a consequence of the definitions:
Proposition 2.11. Let f : X → Y be (K, C)-quasiisometric embedding. If f induces a Morsepreserving map ∂M f : ∂M X → ∂M Y , then ∂M f is a homeomorphism onto its image.
We note that if f : X → Y is a stable embedding, then all geodesics get sent to uniformly Morse
quasigeodesics. Hence the induced map ∂f : ∂M X → ∂M Y is clearly Morse-preserving and:
Corollary 2.12. Let X, Y be proper geodesic metric spaces. If f : X → Y is a stable embedding,
then the induced map ∂f : ∂M X → ∂M Y is a homeomorphism onto its image. Moreover, if f is
the orbit map of a finitely generated group G acting by isometries on Y , then ∂f is G-equivariant.
3. The action of G on ∂M X
In this section, we begin a study of the dynamics of the action of G on ∂M X.
3.1. Definition of the G-action. For the rest of this section, fix G a group acting by isometries
on a proper geodesic metric space X and a base point e ∈ X.
Lemma 3.1. Given any Morse function N and g ∈ G, there exists a Morse function N 0 depending
(N )
only on N and g such that if (xn ), (yn ) ⊂ Xe are asymptotic, then:
(N 0 )

(1) (g · xn ), (g · yn ) ⊂ Xe
and
(N 0 )
(N 0 )
(2) (g · yn ) ∈ Xe
is asymptotic to (g · xn ) ∈ Xe .
Proof. Since G is acting by isometries we see that g and g · xi are connected by a N -Morse geodesic.
(N )
(N 0 )
As in the proof of Proposition 3.15 in [CH16] we show that Xg·e ⊂ Xe
for some N 0 which
depends only on N and d(g · e, e).
To prove the second part, we note that in the proof of Proposition 3.15 in [CH16] we also get
(N )
that there exists a constant D such that for all x, y ∈ ∂Xg·e ,
(x ·N 0 y)e − D ≤ (x ·N y)g·e ≤ (x ·N 0 y)e + D.
(N )

Since G acts by isometries and (xn ), (yn ) ⊂ Xe are asymptotic, it follows that (g · xn ), (g · yn ) ⊂
(N )
(N )
Xg·e are asymptotic, which forces (g · xn ), (g · yn ) ⊂ Xe to be asymptotic.

We may naturally extend the action of G on X to an action on the whole Morse boundary, ∂M X,
(N )
(N )
as follows: Suppose (xn ) ⊂ Xe is a sequence N -converging to a point λ ∈ ∂Xe . For any g ∈ G,
0
(N )
we define g · λ to be the asymptotic class of (g · xn ) ∈ ∂Xe , where N 0 is the Morse gauge from
Lemma 3.1.
3.2. Definition of the limit set.
Definition 3.2 (Λ(G)). The limit set of the G-action on ∂M X is
n
o
Λe (G) = λ ∈ ∂M X ∃N and (gk ) ⊂ G such that (gk · e) ⊂ Xe(N ) and lim gk · e = λ
The following lemma is an immediate consequence of the definitions and Lemma 3.1:
7

Lemma 3.3. For any e, f ∈ X and corresponding natural change of basepoint homeomorphism
φe,f : Λe (G) → Λf (G), we have φe,f (Λe (G)) = Λf (G). Moreover, we have G · Λe (G) ⊂ Λe (G), i.e.,
Λe (G) is G-invariant.
3.3. Limit geodesics and their properties. In Subsection 3.5, we will define the weak hull of
Λe (G) as all geodesics connecting points in Λe (G). For now, however, we will focus our attention
on a special class of visual geodesics arising naturally as limits of finite geodesics whose endpoint
sequences converge to the Morse boundary. These geodesics are easier to work with and more
obviously tied to the geometry of G.
(N )
(N )
Let (xn ), (yn ) ∈ Xe be sequences asymptotic to λ− , λ+ ∈ ∂Xe respectively, with λ− 6= λ+ .
Let γx,n = [e, xn ], γy,n = [e, yn ] be N -Morse geodesics. Since X is proper, the Arzelà-Ascoli theorem
implies that there exist geodesic rays γx , γy such that γx,n , γy,n subsequentially converge uniformly
on compact sets to γx , γy . Let γn be a geodesic joining γx (n) and γy (n) for each n ∈ N.
In fact, one can prove that (γn ) has a Morse subsequential limit:
Lemma 3.4. Using the above notation, there exists a Morse gauge N 0 and a biinfinite N 0 -Morse
geodesic γ : (−∞, ∞) → X such that γn subsequentially converges uniformly on compact sets to γ.
Moreover, we have that γ ⊂ N4N (3,0) (γx ∪ γy ).
Proof. Let R ∈ N be so that d(γx (R), γy (t)) > 4N (3, 0) for all t ∈ [0, ∞). For each natural number
n > R, let γn be the geodesic between γx (n) and γy (n).
By Lemma 2.5, we know that the triangle γx ([0, n]) ∪ γn ∪ γy ([0, n]) is 4N (3, 0)-slim. From the
choice of R, it follows that each γn must intersect a compact ball of radius 4N (3, 0) centered at
γx (R). By Arzelá–Ascoli, there is a subsequence of {γn } which converges to a biinfinite geodesic γ.
Since every γn is in the 4N (3, 0) neighborhood of γx ∪ γy , it follows that γ must be as well. From
this, a standard argument implies that γ((−∞, 0]), γ([0, ∞)) are a bounded Hausdorff distance from
γx , γy , respectively, where that bound depends on N .
Finally, it follows from the moreover statement of Lemma 2.5 that γ is N 0 -Morse, where N 0
depends on N because each γn is N 0 -Morse.

(N )

(N )

Definition 3.5 (Limit geodesics and triangles). Given (xn ), (yn ) ∈ Xe and λ− , λ+ ∈ ∂Xe as
above, we call γx , γy as described above a limit legs, and γ a limit geodesic for λ− , λ+ . We call the
triangle formed by γx ∪ γy ∪ γ a limit triangle based at e.
The next goal is to prove the following two propositions:
(N )

Proposition 3.6 (Limit triangles are slim). For any Morse gauge N , if (xn ), (yn ) ⊂ Xe
(N )
asymptotic to λ− 6= λ+ ∈ ∂Xe , then any limit triangle is 4N (3, 0)-slim.

are

Proposition 3.7 (Limit geodesics are asymptotic). For any Morse gauge N , there exists a constant
(N )
K 0 > 0 such that if γ, γ 0 are limit geodesics with the same endpoints λ+ , λ− ∈ ∂Xe , then
dHaus (γ, γ 0 ) < K 0 .
Proposition 3.7 is an immediate consequence of Proposition 3.6 and following lemma, the proof
of which appears in the proof of Theorem 2.8 of [CH16]:
(N )

Lemma 3.8 (Limit geodesic rays are asymptotic). For any Morse gauge N and λ ∈ ∂Xe
and γ 0 are N -Morse geodesic rays with endpoint λ, then dHaus (γ, γ 0 ) < 14N (3, 0).

, if γ

Proof of Proposition 3.6. By Lemma 3.4, γ ⊂ N4N (3,0) (γx ∪ γy ). Let w ∈ γy and assume that w
is not within 4N (3, 0) of a point in γx . It suffices to find a point on γ within 4N (3, 0) of w. (A
similar argument works when y is not within 4N (3, 0) of γy .)
8

Let z ∈ γ be the closest point on γ to w. By Lemma 2.5, each γx ([0, n]) ∪ γn ∪ γy [(0, n)] is
4N (3, 0)-slim. Since the γn subsequentially converges on compact sets to γ, it follows that for all
 ≥ 0, we have d(w, z) ≤ 4N (3, 0) + . Thus d(w, z) ≤ 4N (3, 0) and the proposition follows from a
symmetric argument with w ∈ γx .

3.4. Asymptoticity. In this subsection, we study the behavior of geodesics and geodesic rays
(N )
asymptotic to points in ∂Xe .
(N )

Definition 3.9 (Asymptotic rays). Let λ ∈ ∂Xe and let γ : [0, ∞) → X with γ(0) = e be a limit
leg for λ based at e. We say that a geodesic γ 0 : [0, ∞) → X with γ 0 (0) = e is asymptotic to λ if
there exists K > 0 such that
dHaus (γ, γ 0 ) < K.
The following is an immediate consequence of Corollary 2.6 in [Cor15]:
Lemma 3.10. There exists K0 > 0 and Morse gauge N 0 depending only on N such that the
(N )
following holds: For any λ ∈ ∂Xe , if γ, γ 0 : [0, ∞) → X are geodesic rays with γ(0) = γ 0 (0) which
0
are asymptotic to λ, then γ, γ are N 0 -Morse and
dHaus (γ, γ 0 ) < K0 .
Definition 3.11 (Asymptotic, bi-asymptotic). Let γ : (−∞, ∞) → X be a biinfinite geodesic in
(N )
X with γ(0) a closest point to e along γ. Let λ ∈ ∂Xe . We say γ is forward asymptotic to λ if
for any N -Morse geodesic ray γλ : [0, ∞) → X with γλ (0) = e, there exists K > 0 such that
dHaus (γ([0, ∞), γλ ([0, ∞)) < K.
We define backwards asymptotic similarly. If γ is forwards, backwards asymptotic to λ, λ0 , respectively, then we say γ is bi-asymptotic to (λ, λ0 ).
We note that it is an immediate consequence of Proposition 3.6 and Lemma 3.10 that limit
geodesics are bi-asymptotic to their endpoints with a uniform asymptoticity constant.
Lemma 3.12. There exists a K1 > 0 depending only on N such that the following holds: Let
γ+ , γ− be N -Morse geodesic rays with γ+ (0), γ− (0) = e, and let γ be a bi-infinite geodesic such that
both
dHaus (γ− ([0, ∞)), γ((−∞, 0])) < K and dHaus (γ+ ([0, ∞)), γ([0, ∞))) < K
for some constant K > 0. Then the triangle γ− ∪ γ ∪ γ+ is K1 -slim. Furthermore, there exist
S, R ∈ [0, ∞) such that the following holds:
dHaus (γ− ([R, ∞)), γ((−∞, 0])), dHaus (γ+ ([S, ∞)), γ([0, ∞))), d(γ− (R), γ(0)), d(γ+ (S), γ(0)) < K1 .
Proof. Up to reparameterization we may assume that γ(0) is the point on γ closest to e. By
assumption we know that there is a constant K > 0 such that dHaus (γ([0, ∞)), γ+ ) < K. Let
t ∈ [6K, ∞) and let t0 ∈ [0, ∞) be such that γ(t0 ) is the closest point along γ to γ+ (t). We claim
that φ = [e, γ(0)] ∪ γ([0, t0 ]) ∪ [γ(t0 ), γ+ (t)] is a (5, 0)-quasigeodesic.
It suffices to check the standard inequality with vertices in different segments of φ. First suppose
that u ∈ [e, γ(0)] and v ∈ [γ(0), γ(t0 )]; the case that u ∈ [γ(t0 ), γ+ (t)] and v ∈ [γ(0), γ(t)] is similar.
We know that d(u, γ(0)) ≤ d(u, v) because γ(0) is a nearest point to e along γ. Note that
d(γ(0), v) ≤ d(u, v) + d(u, γ(0)) by the triangle inequality. Let dφ (u, v) denote the distance along
φ between u and v. We have:
d(u, v) ≤ dφ (u, v) =d(u, γ(0)) + d(γ(0), v)
≤d(u, v) + (d(u, v) + d(u, γ(0))
≤3d(u, v).
9

γ

γ−

γ(t0 )
γ+ (t)
γ+ (t0 )

γ(0)
γ+
γ+ (S)
e

Figure 3.1. Lemma 3.12
Now assume that u ∈ [e, γ(0)] and v ∈ [γ(t0 ), γ+ (t)].
Let ξ = [u, γ(0)] ∪ γ([0, t0 ]) ∪ [γ(t0 ), v] and denote its arclength by kξk. Since t ∈ [6K, ∞), we
know that d(u, v) ≥ t − 2K ≥ 32 t and d(γ(0), γ(t0 )) = t0 < t + 2K < 2t. Putting these inequalities
together:
d(u, v) ≤ kξk =d(u, γ(0)) + d(γ(0), γ(t0 )) + d(γ(s), v)
9
<t/6 + 2t + t/6 < 3t ≤ d(u, v)
2
Thus φ is a (5, 0)-quasi geodesic.
By Lemma 2.4, it follows that dHaus (φ, [e, γ+ (t)) < N (5, 0). Hence there exist S ≥ 0 and
t0 > t0 − K > 0 such that d(γ(0), γ+ (S)), d(γ(t0 ), γ+ (t0 )) < N (5, 0). Let [γ+ (S), γ(0)], [γ(t0 ), γ+ (t0 )]
be geodesics and φ0 = [γ+ (S), γ(0)]∪[γ(0, t0 )]∪[γ(t0 ), γ+ (t0 )]. Then φ0 is a (1, 2N (5, 0))-quasigeodesic
and Lemma 2.4 implies that dHaus (φ0 , [γ+ (S, t0 )]) < N (1, 2N (5, 0)) and hence
dHaus ([γ(0, t0 )], γ+ (S, t0 )) < 2N (1, 2N (5, 0)) + 2N (5, 0).
Since t0 > t0 − K > 0 depends only on t0 and N , we have that t0 → ∞ as t0 → ∞, and hence
dHaus (γ(0, ∞), γ+ (S, ∞)) < 2N (1, 2N (5, 0)) + 2N (5, 0), as required. A similar argument with γ−
provides R ≥ 0 such that dHaus (γ(0, −∞), γ− (R, ∞)) < 2N (1, 2N (5, 0)) + 2N (5, 0).
We know by construction that d(γ− (R), γ(0)), d(γ+ (S), γ(0)) < N (5, 0), so d(γ− (R), γ+ (S)) <
2N (5, 0). We also know by Lemma 2.5 that the triangle γ− ([0, R]) ∪ [γ− (R), γ+ (S)] ∪ γ+ ([0, S]) is
4N (3, 0)-slim. Thus if we set K1 = max{(4N (3, 0) + 2N (5, 0)), (2N (1, 2N (5, 0)) + N (5, 0))}, then
the triangle γ− ∪ γ ∪ γ+ is K1 -slim. This completes the proof.

The following proposition will allow us to define the weak hull in the next subsection:
Proposition 3.13. There exists K2 > 0 depending only on N such that the following holds: Let
(N )
λ− , λ+ ∈ ∂Xe be distinct points. If γ, γ 0 : (−∞, ∞) → X are geodesics with γ(0), γ 0 (0) closest
points to e along γ, γ 0 , respectively, such that γ, γ 0 are bi-asymptotic to (λ− , λ+ ), then
dHaus (γ, γ 0 ) < K2 .
Proof. First assume γ is a limit geodesic.
Since γ is a limit geodesic, Lemma 2.5 implies that γ is N 0 -Morse where N 0 depends only on N .
By Lemma 3.12, we know that there exist S, R ∈ [0, ∞) such that each of the following holds:
dHaus (γ([R, ∞)), γ 0 ((−∞, 0])) < K1 , dHaus (γ([S, ∞)), γ 0 ([0, ∞))) < K10 ,
and
d(γ(R), γ 0 (0)), d(γ(S), γ 0 (0)) < K10 ,
10

where K10 depends only on N 0 .
We know then that d(γ(R), γ(S)) < 2K10 . Putting together these facts, we get that the Hausdorff
distance between γ, γ 0 is less than 2K10 .
It follows that any two geodesics γ, γ 0 have Hausdorff distance bounded by K2 = 4K10 .

The following is an immediate corollary of Lemma 3.12 and Proposition 3.13:
(N )

Corollary 3.14. There exists K1 such that for any distinct λ+ , λ− ∈ ∂Xe , any geodesic rays
γ+ , γ− asymptotic to λ+ , λ− respectively with γ+ (0) = γ− (0) = e, and any bi-infinite geodesic γ
bi-asymptotic to λ+ , λ− , the triangle γ+ ∪ γ ∪ γ− is K1 -slim.
3.5. The weak hull He (G). We are ready to define the weak hull of G in X.
Definition 3.15. The weak hull of G in X based at e ∈ X, denoted He (G), is the collection of all
biinfinite rays γ which are bi-asymptotic to (λ, λ0 ) for some λ 6= λ0 ∈ Λe (G).
Note that Proposition 3.13 says that any two rays γ, γ 0 ∈ He (G) that are both bi-asymptotic to
(N )
(λ, λ0 ) have bounded Hausdorff distance, where that bound depends on the stratum Xe in which
0
the sequences defining λ, λ live. Also note that the proof of Lemma 3.4 implies that limit geodesics
are in He (G).
The following lemma is evident from the definitions:
Lemma 3.16. If |Λe (G)| ≥ 2, then He (G) is nonempty and G-invariant.
The following is an interesting question:
Question 3.17. If |Λ(G)| =
6 ∅, then must we have in fact |Λ(G)| ≥ 2?
In the case of CAT(0) spaces, this question has been affirmatively answered [Mur15, Lemma 4.9].
4. Boundary convex cocompactness and stability
In this section, we prove the main theorem, Theorem 1.6, namely that boundary convex cocompactness as in Definition 1.4 and stability as in Definition 2.9 are equivalent.
4.1. Compact of limit sets have stable weak hulls. For the rest of this section, fix a group G
acting by isometries on a proper geodesic metric space X.
In this subsection, we will prove that if G has a compact limit set, then He (G) is stable.
Lemma 4.1. If K ⊂ ∂M X is nonempty and compact, then for any e ∈ X there exists N > 0 such
(N )
that K ⊂ ∂Xe .
Proof. We will closely follow the proof of Lemma 3.3 in [Mur15].
(N )
N X [CH16, Theorem 3.14].
Since X is proper, we use the fact that ∂Xe is homeomorphic to ∂M
N X for any Morse gauge N . Then we know that there is a
Assume that K is not contained in ∂M
sequence (αi ) ⊂ K of Ni -Morse geodesics where Ni > Ni−1 + 1 for all i. Let An = {αi }i≥n+1 . We
N X is finite. Since singletons are closed in each ∂ N X, each of
note that for all n and N , An ∩ ∂M
M
which is Hausdorff, it follows that singletons are closed in ∂M X. Thus An is closed in ∂M X by the
definition of the direct limit topology.
The collection {∂M X\An }n∈N is an open cover of K, but each ∂M X \ An only contains a finite
number of the αi , so that any finite subcollection of {∂M X\An }n∈N will only contain finitely many
αi . This contradicts the fact that K is compact, completing the proof.

The following proposition is the main technical statement of this section:
(N )

Proposition 4.2 (Compact limit sets have stable hulls). If Λe (G) ⊂ ∂Xe is compact for some
(any) e ∈ X, then for each e ∈ X there exists a Morse gauge N 0 such that He (G) is N 0 -stable.
11

γx0

γx

x
x00

x0

e
y 00

y0
y

γ0

γy

γy0

Figure 4.1. Proposition 4.2
Proof. Let x, y ∈ He (G). By Lemma 3.4, we may assume that x, y do not lie on the same geodesic.
Let [x, y] be any geodesic between x and y.
Since He (G) is a subspace with the induced metric, it suffices to prove that [x, y] is uniformly
Morse. Moreover, since every hull geodesic lies within uniform Hausdorff distance of a limit geodesic
by Proposition 3.13 and Lemma 4.1, it suffices to consider the case when x and y lie on limit
geodesics by Lemma 2.4.
Let γx , γy be distinct limit geodesics on which x, y lie, respectively. By Proposition 3.6, there
exist limit legs γx0 , γy0 based at e and points x0 ∈ γx0 , y 0 ∈ γy0 such that d(x, x0 ), d(y, y 0 ) ≤ 4N (3, 0).
Let γ 0 be the limit geodesic which forms a limit triangle with limit legs γx0 and γy0 . Note by
definition γ 0 ∈ He (G). By Proposition 3.6, the limit triangle γ 0 ∪ γx0 ∪ γy0 is 4N (3, 0)-thin, so there
exist x00 , y 00 ∈ γ 0 with d(x0 , x00 ), d(y 0 , y 00 ) < 4N (3, 0), and hence d(x, x00 ), d(y, y 00 ) < 8N (3, 0).
Let [x00 , x], [y, y 00 ] be any geodesics between x, x00 and y, y 00 , respectively. Then the concatenation
σ = [x00 , x] ∪ [x, y] ∪ [y, y 00 ] gives a (1, 16N (3, 0))-quasigeodesic with endpoints x00 , y 00 on the N 0 Morse geodesic γ 0 . If [x00 , y 00 ] ⊂ γ 0 is the subsegment of γ 0 between x00 and y 00 , then σ is in the
N 0 (1, 16N (3, 0)) neighborhood of [x00 , y 00 ]. An easy argument then implies that σ is uniformly
Morse, from which it follows immediately that [x, y] is uniformly Morse.

4.2. Boundary convex cocompactness implies stability. We now prove the first direction of
the main theorem:
Theorem 4.3. Suppose G acts by isometries on a proper geodesic metric space X such that
(1) The action of G on X is proper,
(2) Λe (G) is nonempty and compact for any e ∈ X, and
(3) G acts cocompactly on He (G) for any e ∈ X.
Then any orbit of G is a stable subspace of X and the orbit map extends continuously and Gequivariantly to an embedding of ∂Gr G into ∂M Xe which is a homeomorphism onto its image Λe (G).
Proof. Since G acts properly and cocompactly on He (G), it follows that the orbit map g 7→ g · e
is a quasiisometry G → He (G), where we consider He (G) with the metric induced from X. In
particular, G · e is quasiisometrically embedded in X.
12

Let x ∈ He (G) and consider G · x ⊂ He (G) ⊂ X. Let y, z ∈ G · x. By Proposition 4.2,
there exists a Morse gauge N such that any geodesic [y, z] between y, z is N -Morse. Hence, any
(K, C)-quasigeodesic between y, z must stay within N (K, C) of [y, z], implying that G · x is a stable
subspace of X. In particular, G is hyperbolic, so ∂Gr G is defined.
By Corollary 2.12, the orbit map g 7→ g · e induces a topological embedding f : ∂Gr G → ∂M X
which gives a homeomorphism f : ∂Gr G → Λe (G). This completes the proof.

4.3. Stability implies boundary convex cocompactness. Finally, we prove the second direction:
Theorem 4.4. Suppose G acts by isometries on a proper geodesic metric space X such that any
orbit map of G is an infinite diameter stable subspace of X. Then:
(1) Λ(G) is nonempty and compact,
(2) He (G) is a stable subset of X,
(3) G acts cocompactly on He (G), and
(4) Any orbit map extends continuously and G-equivariantly to an embedding of ∂Gr G into
∂M Xe which is a homeomorphism onto its image Λe (G).
Proof. G is hyperbolic because its orbit is stable, so we know that ∂M G = ∂Gr G is compact. By
Corollary 2.12, any orbit map g 7→ g · e induces a homeomorphism ∂Gr G → Λe (G), giving us (1)
and (4).
Thus by Proposition 4.2, the weak hull He (G) is stable in X, proving (3). It remains to prove
that G acts cocompactly on He (G).
To see this, let z ∈ He (G). By Proposition 3.13, we may assume that z ∈ γ for some limit
geodesic γ with ends (gn · e), (hn · e). Let γg , γh be the limit legs of the corresponding limit triangle
for γ. By Lemma 3.6, there exists x ∈ γg ∪ γh such that d(x, z) < 4N (3, 0), where N is the Morse
(N )
gauge such that Λe (G) ⊂ ∂Xe .
Without loss of generality, assume that x ∈ γg . By definition of γg , there exists a sequence of
N -Morse geodesics γn = [e, gn · e] which subsequentially converges to γg uniformly on compact sets.
Hence, by passing to a subsequence if necessary, for any  > 0 and every T > 0, there exists M > 0
such that for any n ≥ M , we have
dHaus (γn ([0, T ]), γg ([0, T ])) < .
Taking T > 0 sufficiently large so that x ∈ γg ([0, T ]) and taking n > M for M corresponding to
this T , we may assume there exists y ∈ γn such that d(y, x) < .
Since G·e is quasiisometrically embedded, there exist K, C such that any geodesic in G between 1
and gn maps to a (K, C)-quasigeodesic qn between e and gn · e. Let qn0 denote the (K, C + (K + C))quasigeodesic obtained by connecting successive vertices of qn by geodesics of length at most K +C.
Since [e, gn · e] is N -Morse, it follows that
there exists w0 ∈ qn0 such that d(w0 , y) < 2N (K, 2C + K), and thus w ∈ qn with d(w, y) <
2N (K, 2C + K) + K+C
2 . It follows that


K +C
d(z, w) ≤ d(z, x) + d(x, y) + d(y, w) < 4N (3, 0) +  + 2N (K, 2C + K) +
2
which is a constant depending only on N and the quasiisometric embedding constants of G in X.
Hence G acts cocompactly on He (G), as required.

References
[ADT16] Tarik Aougab, Matthew Gentry Durham, and Samuel J Taylor, Middle recurrence and pulling back stability
for proper actions, in preparation (2016).
13

[AK11] Yael Algom-Kfir, Strongly contracting geodesics in outer space, Geometry & Topology 15 (2011), no. 4,
2181–2233.
[BC12] J. Behrstock and R. Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann.
352 (2012), 339–356.
[Beh06] J. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geometry & Topology 10 (2006), 2001–2056.
[BH99] Martin R Bridson and André Haefliger, Metric spaces of non-positive curvature, Vol. 319, Springer, 1999.
[CH16] Matthew Cordes and David Hume, Stability and the morse boundary, arXiv preprint arXiv:1606.00129
(2016).
[Cor15] Matthew Cordes, Morse boundaries of proper geodesic metric spaces, arXiv:1502.04376 (2015).
[DHS16] Matthew G Durham, Mark F Hagen, and Alessandro Sisto, Boundaries and automorphisms of hierarchically
hyperbolic spaces, arXiv preprint arXiv:1604.01061 (2016).
[DT14] Spencer Dowdall and Samuel J Taylor, Hyperbolic extensions of free groups, arXiv preprint arXiv:1406.2567
(2014).
[DT15a]
, Contracting orbits in outer space, arXiv preprint arXiv:1502.04053 (2015).
[DT15b] Matthew Durham and Samuel J Taylor, Convex cocompactness and stability in mapping class groups,
Algebraic & Geometric Topology 15 (2015), no. 5, 2839–2859.
[Far98] Benson Farb, Relatively hyperbolic groups, Geometric and functional analysis 8 (1998), no. 5, 810–840.
[FM02] Benson Farb and Lee Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002),
91–152 (electronic). MR1914566
[Hag08] Frédéric Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167–209.
MR2413337
[Ham05] Ursula Hamenstädt, Word hyperbolic extensions of surface groups, arXiv preprint math (2005).
[HH14] Ursula Hamenstädt and Sebastian Hensel, Convex cocompact subgroups of Out(f), arXiv preprint
arXiv:1411.2281 (2014).
[KL08] Richard P. Kent IV and Christopher J. Leininger, Shadows of mapping class groups: capturing convex
cocompactness, Geom. Funct. Anal. 18 (2008), no. 4, 1270–1325. MR2465691
[KMT14] Thomas Koberda, Johanna Mangahas, and Samuel J Taylor, The geometry of purely loxodromic subgroups
of right-angled artin groups, To appear in the Transactions of the American Mathematical Society; arXiv
preprint arXiv:1412.3663 (2014).
[Mar74] Albert Marden, The geometry of finitely generated kleinian groups, Annals of Mathematics (1974), 383–462.
[Mur15] Devin Murray, Topology and dynamics of the contracting boundary of cocompact cat (0) spaces, arXiv
preprint arXiv:1509.09314 (2015).
[Sul85] Dennis Sullivan, Quasiconformal homeomorphisms and dynamics i. solution of the fatou-julia problem on
wandering domains, Annals of mathematics 122 (1985), no. 2, 401–418.
[Swe01] Eric L Swenson, Quasi-convex groups of isometries of negatively curved spaces, Topology and its Applications 110 (2001), no. 1, 119–129.
[TT16] Samuel J Taylor and Giulio Tiozzo, Random extensions of free groups and surface groups are hyperbolic,
International Mathematics Research Notices 2016 (2016), no. 1, 294–310.
Department of Mathematics, Brandeis University, 415 South Street, Waltham, MA 02453, U.S.A.
E-mail address: mcordes@brandeis.edu
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48105,
U.S.A.
E-mail address: durhamma@umich.edu

14

