arXiv:1708.07818v2 [] 1 Sep 2017

ON BOUNDARIES OF RELATIVELY HYPERBOLIC
RIGHT-ANGLED COXETER GROUPS
MATTHEW HAULMARK, HOANG THANH NGUYEN, AND HUNG CONG TRAN
Abstract. We give “visual descriptions” of cut points and non-parabolic
cut pairs in the Bowditch boundary of relatively hyperbolic right-angled
Coxeter groups. We also prove necessary and sufficient conditions for a
relatively hyperbolic right-angled Coxeter group whose defining graph
has a planar flag complex with minimal peripheral structure to have
the Sierpinski carpet or the 2-sphere S2 as its Bowditch boundary. We
apply these results to the problem of quasi-isometry classification of
right-angled Coxeter groups. Additionally, we study right-angled Coxeter groups with isolated flats whose CAT(0) boundaries are Menger
curve.

1. Introduction
For each finite simplicial graph Γ the associated right-angled Coxeter
group GΓ has generating set S equal to the vertices of Γ, relations s2 = 1 for
each s in S and relations st = ts whenever s and t are adjacent vertices. In
geometric group theory, groups acting on CAT(0) cube complexes are fundamental objects and right-angled Coxeter groups provide a rich source of
these such groups. The coarse geometry of right-angled Coxeter groups was
studied by Caprace [Cap09, Cap15], Dani-Thomas [DT15, DT], Dani-StarkThomas [DST], Behrstock-Hagen-Sisto [BHS17], Levcovitz [Lev] and others.
In this paper, we will study boundaries of relatively hyperbolic right-angled
Coxeter groups.
The notion of a relatively hyperbolic group was introduced by Gromov
[Gro87] to generalize both word hyperbolic and geometrically finite Kleinian
groups. Introduced by Bowditch [Bow12] there is a boundary for relatively
hyperbolic groups. The Bowditch boundary generalizes the Gromov boundary of a word hyperbolic group and the limit set of a geometrically finite
Kleinian group. Under modest hypotheses on the peripheral subgroups, the
homeomorphism type of the Bowditch boundary is known to be a quasiisometry invariant of the group (see Groff [Gro13]). Combining the work of
Groff [Gro13] and Behrstock-Druţu-Mosher [BDM09], we elaborate on the
homeomorphism between Bowditch boundaries induced by a quasi-isometry
between two relatively hyperbolic groups whose peripheral subgroups are
not relatively hyperbolic (see Theorem 2.12).
Date: September 4, 2017.
2000 Mathematics Subject Classification. 20F67, 20F65.
1

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In [Cap09, Cap15], Caprace gives necessary and sufficient conditions on
the defining graph Γ for the associated right-angled Coxeter group GΓ to
be relatively hyperbolic with respect to a collection of finitely generated
subgroups. Behrstock-Hagen-Sisto [BHS17] develop further the work of
Caprace to describe the minimal peripheral structure of a relatively hyperbolic right-angled Coxeter group (see Definition 2.8). In Sections 3 and 4,
we study Bowditch boundaries of relatively hyperbolic right-angled Coxeter
groups with minimal peripheral structure. We use our results concerning the
Bowditch boundary to study quasi-isometry classification of certain classes
of right-angled Coxeter groups. Lastly, we will also study CAT(0) boundaries of right-angled Coxeter groups with isolated flats.
1.1. Visual descriptions of cut points and non-parabolic cut pairs.
Cut points and non-parabolic cut pairs are topological features of the Bowditch
boundary of a relatively hyperbolic group. In relatively hyperbolic rightangled Coxeter groups, we are able to visualize these features via defining
graphs. Our first main result of Section 3 is the following theorem relating
cut points in the Bowditch boundary of a relatively hyperbolic right-angled
Coxeter group to the defining graph.
Theorem 1.1. Let Γ be a simplicial graph and J be a collection of induced
proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ
is one-ended, hyperbolic relative to the collection P = { GJ | J ∈ J }, and
suppose each subgroup in P is also one-ended. Then each parabolic point
vgGJ0 is a global cut point if and only if some induced subgraph of J0 separates
the graph Γ.
Therefore, we obtain a new quasi-isometry invariant among all relatively
hyperbolic right-angled Coxeter groups.
Corollary 1.2. Let Γ1 and Γ2 be simplicial graphs such that GΓ1 and GΓ2
are both one-ended. Assume that each graph Γi has a peripheral structure
Ji that consists of proper subgraphs of Γi such that each subgroup in Pi =
{ GJ | J ∈ Ji } is one-ended and not relatively hyperbolic. If GΓ1 and GΓ2
are quasi-isometric, then for each graph K ∈ J1 there is a graph L ∈ J2 such
that GK and GL are quasi-isometric and vice versa. Moreover, if K has
an induced subgraph that separates Γ1 , then L also has an induced subgraph
which separates Γ2 .
We refer the reader to Example 3.7 for an illustration of the application of Theorem 1.1 to the quasi-isometry classification of right-angled Coxeter groups. In Section 3 we also visualize non-parabolic cut pairs of the
Bowditch boundary of a relatively hyperbolic group via its defining graph.
Theorem 1.3. Let Γ be a simplicial graph and J be a collection of induced
proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is
one-ended and hyperbolic relative to the collection P = { GJ | J ∈ J }, and
suppose each subgroup in P is one-ended. If the Bowditch boundary ∂(GΓ , P)

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

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has a non-parabolic cut pair, then Γ has a separating complete subgraph
suspension. Moreover, if Γ has a separating complete subgraph suspension
whose non-adjacent vertices do not lie in the same subgraph J ∈ J, then the
Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair.
We remark that Theorem 1.3 can help us prove that Bowditch boundaries
of two distinct groups GΓ1 and GΓ2 are not homeomorphic in certain cases
by detecting non-parabolic cut pairs in their Bowditch boundaries. Therefore, we can conclude these groups are not quasi-isometric if their Bowditch
boundaries are based on their minimal peripheral structures (see Example
3.10).
1.2. Relatively hyperbolic right-angled Coxeter groups with Sierpinski carpet or sphere Bowditch boundary. In Section 4, we also give
necessary and sufficient conditions for a relatively hyperbolic right-angled
Coxeter group whose defining graph has planar flag complex to have the
Sierpinski carpet or S2 as its Bowditch boundary. We refer the reader to
the beginning of Section 4 for definitions of the graph theoretic terms used
in the statement of the following theorem.
Theorem 1.4. Let Γ be a graph whose flag complex is planar. Assume
that Γ has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }.
The Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet if and only
if Γ is inseparable and each graph in J is a strongly non-separating 4-cycle
extension graph.
In addition, the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if
and only if Γ contains a strongly non-separating induced n–cycle extension
K (n ≥ 5) satisfying the following properties:
(1) K does not contain any nonadjacent vertices of an induced 4-cycle
extension;
(2) No vertex outside K is adjacent to two non-adjacent vertices of K.
A graph is inseparable if it is connected, has no separating complete subgraph, no cut pair, and no separating complete subgraph suspension. We
note that if the defining graph Γ is inseparable, distinct from a complete
graph, and has planar flag complex distinct from a triangulation of S 2 as in
Theorem 1.4, then the CAT(0) boundary of a Davis complex ΣΓ is a Sierpinski carpet (see Świa̧tkowski [Ś]). With the additional assumption that
Γ has a non-trivial peripheral structure, the group GΓ becomes a relatively
hyperbolic group and Theorem 1.4 gives a classification of such right-angled
Coxeter groups. More importantly, the classification in Theorem 1.4 also
contributes to the problem of quasi-isometry classification of right-angled
Coxeter groups as we demonstrate in Example 4.15.
We now restrict Theorem 1.4 to the case of 2–dimensional right-angled
Coxeter groups GΓ (i.e. Γ is triangle free and has at least one edge) and

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

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we obtain a slightly simpler characterization of relatively hyperbolic rightangled Coxeter groups whose boundaries are the Sierpinski carpet or the
sphere.
Corollary 1.5. Let Γ be a triangle free, planar graph. Assume that Γ has a
non-trivial peripheral structure J and let P = { GJ | J ∈ J }. The Bowditch
boundary ∂(GΓ , P) is S2 or the Sierpinski carpet if and only if Γ is inseparable
and each graph in J is a strongly non-separating 4-cycle.
In addition, the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if
and only if Γ contains a strongly non-separating induced n–cycle K (n ≥ 5)
satisfying the following properties:
(1) K does not contain any nonadjacent vertices of an induced 4-cycle;
(2) No vertex outside K is adjacent to two non-adjacent vertices of K.
We remark that in [DT] Dani-Thomas study 2–dimensional hyperbolic
right-angled Coxeter groups. The above corollary can be considered an
extension of their work to relatively hyperbolic right-angled Coxeter groups.
We also note that the above corollary is no longer true if we drop the planar
condition on the defining graph (see Remark 5.9).
1.3. Non-hyperbolic right-angled Coxeter groups with Menger curve
boundary. Hyperbolic groups with 1-dimensional boundary, generically
have a boundary which is homeomorphic to the Menger curve [KK00, DGP11].
For non-hyperbolic groups, examples of groups with Menger curve boundary are scarce. In fact, prior to [Haua] there were no known techniques for
constructing non-hyperbolic groups with Menger curve CAT(0) boundary.
The first example of a non-hyperbolic group with Menger curve boundary
is constructed in [DHW] using 3-manifolds. In Section 5 we use the main
result of [Haua] and a theorem of Mihalik-Tschantz [MT09] to prove the
following:
Proposition 1.6. Let Γ be a triangle free, non-planar graph with a nontrivial peripheral structure J that consists of induced 4-cycles. If the graph
Γ is inseparable and the CAT(0) boundary ∂ΣΓ of the right-angled Coxeter
Davis complex ΣΓ is not a Sierpinski carpet, then ∂ΣΓ is a Menger curve.
Inspired by the examples in Dani-Haulmark-Walsh [DHW], we use Proposition 1.6 to construct examples of non-hyperbolic right-angled Coxeter
groups with Menger curve CAT(0) boundary. We then use the Bowditch
boundary to show that the constructed examples are not quasi-isometric
(see Lemma 5.7).
Acknowledgments. First, all three authors would like to thank Chris
Hruska for suggestions and insights. The first author would also like thank
Genevieve Walsh for many helpful conversations. Lastly, the authors would
like to thank Jason Behrstock for a correction to Corollary 1.2.

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

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2. Preliminaries
In this section, we review some concepts in geometric group theory:
CAT(0) spaces, δ–hyperbolic spaces, CAT(0) spaces with isolated flats, relatively hyperbolic groups, CAT(0) boundaries, Gromov boundaries, Bowditch
boundaries, and peripheral splitting of relatively hyperbolic groups. We
also use the work of Behrstock-Druţu-Mosher [BDM09] and Groff [Gro13]
to prove that Bowditch boundary is a quasi-isometry invariant among relatively hyperbolic groups with non-relatively hyperbolic peripheral subgroup
structures. We review right-angled Coxeter groups and discuss the work of
Caprace [Cap09, Cap15] and Behrstock-Hagen-Sisto [BHS17] on peripheral
structures of relatively hyperbolic right-angled Coxeter groups.
2.1. CAT(0) spaces, δ–hyperbolic spaces, and relatively hyperbolic
groups. We first discuss on CAT(0) spaces, δ–hyperbolic spaces, Gromov
boundaries, and CAT(0) boundaries. We refer the reader to the book [BH99]
for more details.
Definition 2.1. We say that a geodesic triangle ∆ in a geodesic space X
satisfies the CAT (0) inequality if d(x, y) ≤ d(x, y) for all points x, y on the
edges of ∆ and the corresponding points x, y on the edges of the comparison
triangle ∆ in Euclidean space E2 .
Definition 2.2. A geodesic space X is said to be a CAT (0) space if every
triangle in X satisfies the CAT(0) inequality.
If X is a CAT(0) space, then the CAT (0) boundary of X, denoted ∂X, is
defined to be the set of all equivalence classes of geodesic rays in X, where
two rays c and c′ are equivalent if the Hausdorff distance between them is
finite.
We note that for any x ∈ X and ξ ∈ ∂X there is a unique geodesic ray
αx,ξ : [0, ∞) → X with αx,ξ = x and [αx,ξ ] = ξ. The CAT(0) boundary has
′
a natural topology with
 basis given by the sets U (x, ξ, R, ǫ) = { ξ ∈ ∂X |
d αx,ξ (R), αx,ξ ′ (R) ≤ ǫ }, where x ∈ X, ξ ∈ ∂X, R > 0 and ǫ > 0.
Definition 2.3. A geodesic metric space (X, d) is δ–hyperbolic if every geodesic triangle with vertices in X is δ–thin in the sense that each side lies
in the δ–neighborhood of the union of other sides. If X is a δ–hyperbolic
space, then we could build the Gromov boundary of X, denoted ∂X, in the
same way as for a CAT(0) space. That is, the Gromov boundary of X is
defined to be the set of all equivalence classes of geodesic rays in X, where
two rays c and c′ are equivalent if the Hausdorff distance between them is
finite. However, the topology on it is slightly different from the topology on
the boundary of a CAT(0) space (see for example [BH99, Section III.3] for
details).
We now review relatively hyperbolic groups and related concepts.

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Definition 2.4 (Combinatorial horoball [GM08]). Let T be any graph with
the vertex set V . We define the combinatorial horoball based at T , H(=
H(T )) to be the following graph:
(1) H(0) = V × {{0} ∪ N}.
(2) H(1) = {((t, n), (t, n + 1))} ∪ { ((t1 , n), (t2 , n)) | dT (t1 , t2 ) ≤ 2n }. We
call edges of the first set vertical and of the second horizontal.
Remark 2.5. In [GM08], the combinatorial horoball is described as a 2complex, but we will only require we only require the 1-skeleton for the
horoball in this paper.
Definition 2.6 ([GM08]). Let H be the horoball based at some graph T .
Let D : H → [0, ∞) be defined by extending the map on vertices (t, n) → n
linearly across edges. We call D the depth function for H and refer to
vertices v with D(v) = n as vertices of depth n or depth n vertices.
Because T × {0} is homeomorphic to T , we identify T with D −1 (0).
Definition 2.7 (Cusped space [GM08]). Let G be a finitely generated group
and P a finite collection of finitely generated subgroups of G. Let S be a
finite generating set of G such that S ∩ P generates P for each P ∈ P. For
each left coset gP of subgroup P ∈ P let H(gP ) be the horoball based at a
copy of the subgraph TgP with vertex set gP of the Cayley graph Γ(G, S).
The cusped space X(G, P, S) is the union of Γ(G, S) with H(gP ) for every
left coset of P ∈ P, identifying the subgraph TgP with the depth 0 subset
of H(gP ). We suppress mention of S and P when they are clear from the
context.
Definition 2.8 (Relatively hyperbolic group [GM08]). Let G be a finitely
generated group and P a finite collection of finitely generated proper subgroups of G. Let S be a finite generating set of G such that S ∩ P generates
P for each P ∈ P. If the cusped space X(G, P, S) is δ–hyperbolic then we
say that G is hyperbolic relative to P or that (G, P) is a relatively hyperbolic.
Collection P is a peripheral structure, each group P ∈ P is a peripheral subgroup and its left cosets are peripheral left cosets. The peripheral structure
P is minimal if for any other peripheral structure Q on G, each P ∈ P is
conjugate into some Q ∈ Q.
Remark 2.9. Replacing S for some other finite generating set S ′ may
change the value of δ, but does not affect the hyperbolicity of the cusped
space for some δ′ (see [GM08]). Consequently, the concept of relatively
hyperbolic group does not depend on the choice of finite generating set.
We say that a finitely generated group is not relatively hyperbolic if it is
not relatively hyperbolic with respect to any collection of proper subgroups.
2.2. The Bowditch boundary. We now discuss the Bowditch boundary of
a relatively hyperbolic group and prove that it is a quasi-isometry invariant.
We also recall peripheral splitting of relatively hyperbolic groups.

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Definition 2.10 (Bowditch boundary [Bow12]). Let (G, P) be a finitely
generated relatively hyperbolic group. Let S be a finite generating set of
G such that S ∩ P generates P for each P ∈ P. The Bowditch boundary,
denoted ∂(G, P), is the Gromov boundary of the associated cusped space,
X(G, P, S).
Remark 2.11. There is a natural topological action of G on the Bowditch
boundary ∂(G, P) that satisfies certain properties (see [Bow12]).
Bowditch has shown that the Bowditch boundary does not depend on the
choice of finite generating set (see [Bow12]). More precisely, if S and T are finite generating sets for G as in the above definition, then the Gromov boundaries of the cusped spaces X(G, P, S) and X(G, P, T ) are G–equivariantly
homeomorphic (see [Bow12]).
For each peripheral left coset gP the limit set of the associated horoball
H(gP ) consists of a single point in ∂(G, P), called parabolic point vgP . The
stabilizer of the point vgP is the subgroup gP g−1 . We call each infinite
subgroup of gP g−1 a parabolic subgroup and subgroup gP g−1 a maximal
parabolic subgroup.
The homeomorphism type of the Bowditch boundary was already known
to be a quasi-isometry invariant of the group (see Groff [Gro13]) under modest hypotheses on the peripheral subgroups. However, we combine the work
of Groff [Gro13] and Behrstock-Druţu-Mosher [BDM09] to elaborate the
homeomorphism between Bowditch boundaries induced by a quasi-isometry
between two relatively hyperbolic groups whose peripheral subgroups are
not relatively hyperbolic.
Theorem 2.12. Let (G1 , P1 ) and (G2 , P2 ) be finitely generated relatively
hyperbolic groups such that all peripheral subgroups of both (G1 , P1 ) and
(G2 , P2 ) are not relatively hyperbolic. If G1 and G2 are quasi-isometric,
then there is a homeomorphism f from ∂(G1 , P1 ) to ∂(G2 , P2 ) that maps
the set of parabolic points of ∂(G1 , P1 ) bijectively onto the set of parabolic
points of ∂(G2 , P2 ). Moreover, if a parabolic point v of ∂(G1 , P1 ) is labelled
by some peripheral left coset g1 P1 in G1 and the parabolic point f (v) of
∂(G2 , P2 ) is labelled by some peripheral left coset g2 P2 in G2 , then P1 and
P2 are quasi-isometric.
Proof. Fix generating sets S1 and S2 as in Definition 2.10 for G1 and G2
respectively, then there is a quasi-isometry q : Γ(G1 , S1 ) → Γ(G2 , S2 ). By
Theorem 4.1 in [BDM09], the map q takes a peripheral left coset g1 P1 of G1
to within a uniform bounded distance of the corresponding peripheral left
coset g2 P2 of G2 . In particular, P1 and P2 are quasi-isometric. Using the
proof of Theorems 6.3 in [Gro13], we can extend q to the quasi-isometry q̂ :
X(G1 , P1 , S1 ) → X(G2 , P2 , S2 ) between cusped spaces such that q̂ restricts
to a quasi-isometry embedding on each individual horoball of X(G1 , P1 , S1 )
and the image of the horoball lies in some neighborhood of a horoball of
X(G2 , P2 , S2 ). Therefore, there is a homeomorphism f induced by q̂ from

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

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∂(G1 , P1 ) to ∂(G2 , P2 ) that maps the set of parabolic points of ∂(G1 , P1 )
bijectively onto the set of parabolic points of ∂(G2 , P2 ). Moreover, if a
parabolic point v of ∂(G1 , P1 ) is labelled by some peripheral left coset g1 P1
in G1 and the parabolic point f (v) of ∂(G2 , P2 ) is labelled by some peripheral
left coset g2 P2 in G2 , then by the above observation P1 and P2 are quasiisometric.

Definition 2.13 ([Bow01]). Let G be a group. By a splitting of G, over
a given class of subgroups, we mean a presentation of G as a finite graph
of groups, where each edge group belongs to this class. Such a splitting is
said to be relative to another class P of subgroups if each element of P is
conjugate into one of the vertex groups. A splitting is said to be trivial if
there exists a vertex group equal to G.
Assume G is hyperbolic to a collection P. A peripheral splitting of (G, P)
is a representation of G as a finite bipartite graph of groups, where P consists
precisely of the (conjugacy classes of) vertex groups of one color. Obviously,
any peripheral splitting of (G, P) is relative to P and over subgroups of
elements of P. Peripheral splittings of (G, P) are closely related to cut points
in the Bowditch boundary ∂(G, P) ([Bow01]).
Definition 2.14. Given a compact connected metric space X, a point x ∈ X
is a global cut point (or just simply cut point) if X − {x} is not connected.
If {a, b} ⊂ X contains no cut points and X − {a, b} is not connected, then
{a, b} is a cut pair. A point x ∈ X is a local cut point if X − {x} is not
connected, or X − {x} is connected and has more than one end.
2.3. CAT(0) spaces with isolated flats. In this section, we discuss the
work of Hruska-Kleiner [HK05] on CAT(0) spaces with isolated flats.
Definition 2.15. A k–flat in a CAT(0) space X is an isometrically embedded copy of Euclidean space Ek for some k ≥ 2. In particular, note that a
geodesic line is not considered to be a flat.
Definition 2.16. Let X be a CAT(0) space, G a group acting geometrically
on X, and F a G–invariant set of flats in X. We say that X has isolated
flats with respect to F if the following two conditions hold.
(1) There is a constant D such that every flat F ⊂ X lies in a D–
neighborhood of some F ′ ∈ F.
(2) For each positive r < ∞ there is a constant ρ = ρ(r) < ∞ so that
 for
any two distinct flats F, F ′ ∈ F we have diam Nr (F ) ∩ Nr (F ) < ρ.
We say X has isolated flats if it has isolated flats with respect to some
G–invariant set of flats.
Theorem 2.17 ([HK05]). Suppose X has isolated flats with respect to F.
For each F ∈ F the stabilizer StabG (F ) is virtually abelian and acts cocompactly on F . The set of stabilizers of flats F ∈ F is precisely the set
of maximal virtually abelian subgroups of G of rank at least two. These
stabilizers lie in only finitely many conjugacy classes

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Theorem 2.18 ([HK05]). Let X have isolated flats with respect to F. Then
G is relatively hyperbolic with respect to the collection of all maximal virtually
abelian subgroups of rank at least two.
The previous theorem also has the following converse.
Theorem 2.19 ([HK05]). Let G be a group acting geometrically on a CAT(0)
space X. Suppose G is relatively hyperbolic with respect to a family of virtually abelian subgroups. Then X has isolated flats
A group G that admits an action on a CAT(0) space with isolated flats has
a “well-defined” CAT(0) boundary, often denoted by ∂G, by the following
theorem.
Theorem 2.20 ([HK05]). Let G act properly, cocompactly, and isometrically on CAT(0) spaces X and Y . If X has isolated flats, then so does Y ,
and there is a G–equivariant homeomorphism ∂X → ∂Y .
2.4. Right-angled Coxeter groups and their relatively hyperbolic
structures. In this section, we review the concepts of right-angled Coxeter groups and Davis complexes. We also review the work of Caprace
[Cap09, Cap15] and Behrstock-Hagen-Sisto [BHS17] on peripheral structures of relatively hyperbolic right-angled Coxeter groups.
Definition 2.21. Given a finite simplicial graph Γ, the associated rightangled Coxeter group GΓ is generated by the set S of vertices of Γ and has
relations s2 = 1 for all s in S and st = ts whenever s and t are adjacent
vertices.
Let S1 be a subset of S. The subgroup of GΓ generated by S1 is a rightangled Coxeter group GΓ1 , where Γ1 is the induced subgraph of Γ with
vertex set S1 (i.e. Γ1 is the union of all edges of Γ with both endpoints in
S1 ). The subgroup GΓ1 is called a special subgroup of GΓ .
Definition 2.22. Given a finite simplicial graph Γ, the associated Davis
complex ΣΓ is a cube complex constructed as follows. For every k–clique,
T ⊂ Γ, the special subgroup GT is isomorphic to the direct product of k
copies of Z2 . Hence, the Cayley graph of GT is isomorphic to the 1–skeleton
of a k–cube. The Davis complex ΣΓ has 1–skeleton the Cayley graph of GΓ ,
where edges are given unit length. Additionally, for each k–clique, T ⊂ Γ,
and coset gGT , we glue a unit k–cube to gGT ⊂ ΣΓ . The Davis complex ΣΓ
is a CAT(0) space and the group GΓ acts properly and cocompactly on the
Davis complex ΣΓ (see [Dav08]).
Theorem 2.23 (Theorem A’ in [Cap09, Cap15]). Let Γ be a simplicial
graph and J be a collection of induced subgraphs of Γ. Then the right-angled
Coxeter groups GΓ is hyperbolic relative to the collection P = { GJ | J ∈ J }
if and only if the following three conditions hold:
(1) If σ is an induced 4-cycle of Γ, then σ is an induced 4-cycle of some
J ∈ J.

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(2) For all J1 , J2 in J with J1 6= J2 , the intersection J1 ∩ J2 is empty or
J1 ∩ J2 is a complete subgraph of Γ.
(3) If a vertex s commutes with two non-adjacent vertices of some J in
J, then s lies in J.
Theorem 2.24 (Theorem B in [Cap09, Cap15]). Let Γ be a simplicial graph.
If GΓ is relatively hyperbolic with respect to finitely generated subgroups
H1 , · · · , Hm , then each Hi is conjugate to a special subgroup of GΓ .
Theorem 2.25 (Theorem I in [BHS17]). Let T be the class consisting of the
finite simplicial graphs Λ such that GΛ is strongly algebraically thick. Then
for any finite simplicial graph Γ either: Γ ∈ T , or there exists a collection J
of induced subgraphs of Γ such that J ⊂ T and GΓ is hyperbolic relative to
the collection P = { GJ | J ∈ J } and this peripheral structure is minimal.
Remark 2.26. In Theorem 2.25 we use the notion of strong algebraic thickness which is introduced in [BD14] and is a sufficient condition for a group
to be non-hyperbolic relative to any collection of proper subgroups. We
refer the reader to [BD14] for more details. The following theorem from
[BHS17] characterizes all strongly algebraically thick right-angled Coxeter
groups and it will prove useful for studying peripheral subgroups of relatively
hyperbolic right-angled Coxeter groups.
Theorem 2.27 (Theorem II in [BHS17]). Let T be the class of finite simplicial graphs whose corresponding right-angled Coxeter groups are strongly
algebraically thick. Then T is the smallest class of graphs satisfying the
following conditions:
(1) The 4-cycle lies in T .
(2) Let Γ ∈ T and let Λ ⊂ Γ be an induced subgraph which is not a
complete graph. Then the graph obtained from Γ by coning off Λ is
in T .
(3) Let Γ1 , Γ2 ∈ T and suppose there exists a graph Γ, which is not a
complete graph, and which arises as a subgraph of each of the Γi .
Then the union Λ of Γ1 , Γ2 along Γ is in T , and so is any graph
obtained from Λ by adding any collection of edges joining vertices in
Γ1 − Γ to vertices of Γ2 − Γ.
Definition 2.28. The graphs in the collection T from Theorems 2.25 and
2.27 are called thick graphs. Let Γ be a simplicial, non-thick graph. The collection J of induced subgraphs of Γ from Theorem 2.25 is called a peripheral
structure of Γ and each graph in J is called a peripheral subgraph of Γ. A
peripheral structure J of a graph Γ is non-trivial if J is non-empty and does
not contain Γ. A graph Γ has a non-trivial peripheral structure if and only
if GΓ is non-trivially relative hyperbolic group but GΓ is not hyperbolic.

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3. Bowditch boundaries of relatively hyperbolic right-angled
Coxeter groups
In this section, we give “visual” descriptions of cut points and nonparabolic cut pairs of Bowditch boundaries of relatively hyperbolic rightangled Coxeter groups. The Bowditch boundary is a quasi-isometry invariant. So, these results can be applied, in certain cases, to differentiate two
relatively hyperbolic RACGs in terms of quasi-isometry equivalence.
In [Tra13], the third author investigates the connection between the Bowditch
boundary of a relatively hyperbolic group (G, P) and the boundary of a
CAT(0) space X on which G acts geometrically. For relatively hyperbolic
right-angled Coxeter groups, the relevant result from [Tra13] can be stated
as follows:
Theorem 3.1 (Tran [Tra13]). Let Γ be a finite simplicial graph. Assume
that the right-angled Coxeter group GΓ is relatively hyperbolic with respect
to a collection P of its subgroups. Then the Bowditch boundary ∂(GΓ , P) is
obtained from the CAT(0) boundary ∂ΣΓ by identifying the limit set of each
peripheral left coset to a point. Moreover, this quotient map is GΓ -invariant.
We now introduce some definitions concerning defining graphs of rightangled Coxeter groups that we will use to visualize cut points and nonparabolic cut points in the Bowditch boundary.
Definition 3.2. Let Γ1 and Γ2 be two graphs, the join of Γ1 and Γ2 , denoted
Γ1 ∗Γ2 , is the graph obtained by connecting every vertex of Γ1 to every vertex
of Γ2 by an edge. If Γ2 consists of distinct vertices u and v, then the join
Γ1 ∗ {u, v} is the suspension of Γ1 .
Definition 3.3. Let Γ be a simplicial graph. A pair of non-adjacent vertices
{a, b} in Γ is called a cut pair if {a, b} separates Γ. An induced subgraph Γ1
of Γ is a complete subgraph suspension if Γ1 is a suspension of a complete
subgraph σ of Γ. If σ is a single vertex, then Γ1 is a vertex suspension.
An induced subgraph Γ1 of Γ is separating if Γ1 separates Γ. By this way,
we can also consider a cut pair as a separating complete subgraph suspension
which is a suspension of the empty graph.
We will need the following lemma in order to visualize cut points in the
Bowditch boundary of a relatively hyperbolic right-angled Coxeter group.
Lemma 3.4. Let Γ be a simplicial graph and J a collection of induced proper
subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is oneended, hyperbolic relative to the collection P = { GJ | J ∈ J }, and suppose
each subgroup in P is one-ended. Let J0 be an element in J such that some
induced subgraph of J0 separates the graph Γ. Then we can write Γ = Γ1 ∪Γ2
such that the following conditions hold:
(1) Γ1 , Γ2 are both proper induced subgraphs of Γ;
(2) Γ1 ∩ Γ2 is an induced subgraph of J0 .

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

12

(3) For each J in J, J lies completely inside either Γ1 or Γ2
Proof. Since each subgroup in P is one-ended, each graph J in J is connected and J has no separating complete subgraph. Let L be an induced
subgraph of J0 that separates the graph Γ. Let Γ′1 be L together with some
of the components of Γ − L, and let Γ′2 be L together with the remaining
components of Γ − L. Then, Γ′1 , Γ′2 are both proper induced subgraphs of
Γ, L = Γ′1 ∩ Γ′2 , and Γ = Γ′1 ∪ Γ′2 . Since J0 is a proper subgraph of Γ,
Γ′1 − J0 6= ∅ or Γ′2 − J0 6= ∅ (say Γ′1 − J0 6= ∅).
Let Γ1 = Γ′1 , Γ2 = Γ′2 ∪ J0 . Then, Γ1 is an induced proper subgraph of
Γ. We now prove that Γ2 is also an induced proper subgraph. Choose a
vertex w ∈ Γ1 − J0 = Γ′1 − J0 . Since Γ1 ∩ Γ2 = Γ′1 ∩ (Γ′2 ∪ J0 ) ⊂ J0 , the
vertex w does not belong to Γ2 . Therefore, Γ2 is a proper subgraph of Γ.
We now prove that Γ2 is induced. Let e be an arbitrary edge with endpoints
u and v in Γ2 . If e is an edge of Γ′2 , then e is the edge of Γ2 . Otherwise,
e is an edge of Γ′1 = Γ1 , because Γ = Γ′1 ∪ Γ′2 . In particular, u and v are
also the vertices of Γ1 . Again, Γ1 ∩ Γ2 is a subgraph of J0 . Then u and v
are vertices of J0 . Therefore, e is an edge of J0 , because J0 is an induced
subgraph. Thus, e is also an edge of Γ2 . Thus, Γ2 is an induced subgraph.
This implies that Γ1 ∩ Γ2 is an induced subgraph. We already checked that
Γ1 ∩ Γ2 is a subgraph of J0 , and by construction Γ = Γ1 ∪ Γ2 .
We now prove that for each J in J, J lies completely inside either Γ1 or
Γ2 . By the construction, J0 is a subgraph of Γ2 . Therefore, we only need to
check the case where J 6= J0 . It suffices to show that J lies completely inside
either Γ′1 or Γ′2 . By Theorem 2.23, for each J 6= J0 in J the intersection J ∩J0
is empty or it is a complete subgraph of Γ. Also the intersection J ∩ L is an
induced subgraph of J ∩ J0 if J ∩ L 6= ∅. Therefore, J ∩ L is empty or it
is a complete subgraph of Γ. Recall that each graph in J is connected and
has no separating complete subgraph. Therefore, J − L = J − (J ∩ L) is
connected for each J 6= J0 in J. By the construction J − L lies completely
inside either Γ′1 or Γ′2 . Thus, J also lies completely inside either Γ′1 or Γ′2 .
Therefore, J lies completely inside either Γ1 or Γ2 .

We can now give the visual description of cut points in the Bowditch
boundary of relatively hyperbolic right-angled Coxeter groups.
Theorem 3.5. Let Γ be a simplicial graph and J a collection of induced
proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ
is one-ended, hyperbolic relative to the collection P = { GJ | J ∈ J }, and
suppose each subgroup in P is also one-ended. Then each parabolic point
vgGJ0 is a global cut point if and only if some induced subgraph of J0 separates
the graph Γ.
Proof. We first assume that some induced subgraph of J0 separates the
graph Γ. By Lemma 3.4, we can write Γ = Γ1 ∪ Γ2 such that the following
conditions hold:
(1) Γ1 , Γ2 are both proper induced subgraphs of Γ;

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

13

(2) K = Γ1 ∩ Γ2 is an induced subgraph of J0 .
(3) For each J in J, J lies completely inside either Γ1 or Γ2
This implies that GΓ = GΓ1 ∗GK GΓ2 , GΓ1 6= GΓ , and GΓ2 6= GΓ . Since
J lies completely inside either Γ1 or Γ2 for each J ∈ J, each peripheral
subgroup in P must be a subgroup of GΓ1 or GΓ2 . Therefore, GΓ splits
non-trivially relative to P over the parabolic subgroup GK ≤ GJ0 . By the
claim following Theorem 1.2 of [Bow01] the parabolic point vGJ0 labelled by
GJ0 is a global cut point of ∂(G, P). Also, the group GΓ acts topologically
on ∂(G, P) and gvGJ0 = vgGJ0 . Thus, each parabolic point vgGJ0 is also a
global cut point.
We now assume that some parabolic point vgGJ0 is a global cut point.
Again, the group GΓ acts topologically on ∂(G, P) and gvGJ0 = vgGJ0 .
Therefore, vGJ0 is also a global cut point. So, by Theorem 3.3 of [Haub]
GΓ splits non-trivially over a subgroup H of GJ0 . Theorem 1 of [MT09]
implies that there is some induced subgraph K of Γ which separates Γ such
that GK is contained in some conjugate of H. Therefore, GK is also contained in some conjugate of the peripheral subgroup GJ0 . Moreover, GK
and GJ0 are both special subgroups of GΓ . Thus, K is an induced subgraph
of J0 . (This is a standard fact, and we leave the details to the reader.) 
The following corollary follows directly from Theorems 2.12 and 3.5. This
corollary can be used to distinguish between quasi-isometry classes of certain
relatively hyperbolic right-angled Coxeter groups.
Corollary 3.6. Let Γ1 and Γ2 be simplicial graphs such that GΓ1 and GΓ2
are both one-ended. Assume that each graph Γi has a peripheral structure
Ji that consists of proper subgraphs of Γi such that each subgroup in Pi =
{ GJ | J ∈ Ji } is one-ended and not relatively hyperbolic. If GΓ1 and GΓ2
are quasi-isometric, then for each graph K ∈ J1 there is a graph L ∈ J2
such that GK and GL are quasi-isometric and vice versa. Moreover, if K
has some induced subgraph that separates Γ1 , then L also has some induced
subgraph that separates Γ2 .
We now discuss a few examples related to cut points in Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups. These examples
illustrate an application of Theorem 3.5 to the problem of quasi-isometry
classification of right-angled Coxeter groups.
Example 3.7. Let Γ1 , Γ2 , and Γ3 be the graphs in Figure 1. Observe that
all groups GΓi are one-ended. We will prove that groups GΓ1 , GΓ2 , and GΓ3
are not pairwise quasi-isometric by investigating their peripheral structures.
(1)
(2)
In Γ1 , let K1 and K1 be an induced subgraphs generated by {a1 , a2 , a3 , a4 , a5 }
and {a6 , a7 , a8 , a9 , a10 }, respectively. It is easy to see that Γ1 has only six
(1)
(2)
induced 4-cycles which are not subgraphs of K1 and K1 . Denote these
(i)
(i)
(j)
cycles by L1 (i = 1, 2, · · · , 6). Let J1 be the set of all graphs L1 and K1 .
By Theorems 2.23 and 2.25, J1 is a peripheral structure of Γ1 . Moreover,

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

14

Γ2

Γ1
a2

Γ3
c2

b2
a3

a1

a5

b3

b1

a4

c4

b4
a7

c7

b7
a8

a6

c5

c1

b5

a10

b8

b6

a9

b10

b9

c8

c6
c9

Figure 1. The three groups GΓ1 , GΓ2 , and GΓ3 are pairwise
not quasi-isometric because the Bowditch boundaries with
respect to their minimal peripheral structures are pairwise
not homeomorphic.
no induced subgraph of a graph in J1 separates Γ1 . Therefore by Theorem
3.5, GΓ1 is hyperbolic relative to the collection P1 = { GJ | J ∈ J1 } and the
Bowditch boundary ∂(GΓ1 , P1 ) has no global cut point.
(1)
(2)
Similarly, let K2 and K2 be an induced subgraphs of Γ2 generated by
{b1 , b2 , b3 , b4 , b5 } and {b6 , b7 , b8 , b9 , b10 }, respectively. It is easy to see that Γ2
(i)
has only four induced 4-cycles, denoted L2 (i = 1, 2, · · · , 4), such that each
(1)
(2)
of them is not a subgraph of K2 and K2 . Let J2 be the set of all graphs
(i)
(j)
L2 and K2 . Then by Theorems 2.23 and 2.25, J2 is a peripheral structure
(1)
(2)
of Γ2 . Moreover, K2 and K2 are the only graphs in J2 which contain
induced subgraphs that separate Γ2 . Therefore, GΓ2 is hyperbolic relative
to the collection P2 = { GJ | J ∈ J2 }, the Bowditch boundary ∂(GΓ2 , P2 )
has global cut points and each of them is labelled by some left coset of GK (1)
2
or GK (2) by Theorem 3.5.
2

(1)

Finally, let K3 be an induced subgraph of Γ3 generated by {c6 , c7 , c8 , c9 , c10 }.
(i)
It is easy to see that Γ3 has only three induced 4-cycles, denoted L3
(1)
(i = 1, 2, 3), such that each of them is not a subgraph of K3 . Assume
(1)
that L3 is the induced 4-cycle generated by {c1 , c2 , c5 , c4 }. Let J3 be the
(i)
(1)
set of all graphs L3 and K3 . Again, by Theorem 2.23 and 2.25 we have
(1)
(1)
that J3 is a peripheral structure of Γ3 . Moreover, K3 and L3 are the only
graphs in J3 which contain induced subgraphs that separate Γ3 . Therefore, GΓ3 is hyperbolic relative to the collection P3 = { GJ | J ∈ J3 }, the
Bowditch boundary ∂(GΓ3 , P3 ) has global cut points and each of them is
labelled by some left coset of GK (1) or GL(1) by Theorem 3.5.
3

3

c10

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

15

Note that all the groups in Pi are one-ended. The Bowditch boundary
∂(GΓ1 , P1 ) has no global cut point, but the Bowditch boundaries ∂(GΓ2 , P2 )
and ∂(GΓ3 , P3 ) do. So, GΓ1 cannot be quasi-isometric to GΓ2 and GΓ3 .
Additionally, the Bowditch boundary ∂(GΓ3 , P3 ) has global cut points labelled by some left coset of GL(1) . Meanwhile, no global cut point of the
3

Bowditch boundary ∂(GΓ2 , P2 ) is labelled by the left coset of a peripheral
subgroup that is quasi-isometric to GL(1) . Therefore, GΓ2 and GΓ3 are not
3
quasi-isometric.
In the remainder of this section, we work on the description of nonparabolic cut pairs in Bowditch boundaries of relatively hyperbolic rightangled Coxeter groups in terms of their defining graphs.
Proposition 3.8. Let Γ be a simplicial graph and J be a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter groups
GΓ is one-ended, hyperbolic relative to the collection P = { GJ | J ∈ J },
and suppose each subgroup in P is also one-ended. If Γ has a separating
complete subgraph suspension whose non-adjacent vertices do not lie in the
same subgraph J ∈ J, then the CAT(0) boundary ∂ΣΓ has a cut pair and the
Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair.
Proof. Let K be a separating complete subgraph suspension of Γ whose nonadjacent vertices u and v do not both lie in the same subgraph J ∈ J. Let
T be the set of all vertices of Γ which are both adjacent to u and v. Then T
is a vertex set of a complete subgraph σ of Γ. Otherwise, the two vertices u
and v both lie in the same 4-cycle. Thus, u and v lie in the same subgraph
J ∈ J, a contradiction.
Let K = σ ∗ {u, v}. We can easily verify the following properties of K.
(1) For each J ∈ J the intersection K ∩ J is empty or a complete subgraph.
(2) No vertex outside K is adjacent to the unique pair of nonadjacent
vertices {u, v} of K.
(3) K is an induced subgraph of K.
Therefore, the collection J = J ∪ {K} satisfies all the conditions of Theorem
2.23, which implies that GΓ is hyperbolic relative to the collection P = { GJ |
J ∈ J }.
Using an argument similar to that of Lemma 3.4, we can write Γ = Γ1 ∪Γ2
such that the following conditions hold:
(1) Γ1 , Γ2 are both proper induced subgraphs of Γ;
(2) Γ1 ∩ Γ2 is an induced subgraph L of K.
(3) For each J in J, J lies completely inside either Γ1 or Γ2
Therefore, we can prove that the Bowditch boundary ∂(G, P) has a global
cut point vGK stabilized by the subgroup GK by using an argument similar
to the one in Theorem 3.5.

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

16

By Theorem 3.1, the Bowditch boundary ∂(GΓ , P) is obtained from the
CAT(0) boundary ∂ΣΓ by identifying the limit set of each peripheral left
coset of a subgroup in P to a point. Let f be this quotient map. Since GK is
two-ended, its limit set consists of two points w
 1 and w2 in ∂ΣΓ . Therefore,
f (w1 ) = f (w2 ) = vGK and f ∂ΣΓ − {w1 , w2 } = ∂(GΓ , P) − {vGK }. Since
∂(GΓ , P) − {vGK } is not connected, the space ∂ΣΓ − {w1 , w2 } is also not
connected. This implies that {w1 , w2 } is a cut pair of the CAT(0) boundary
∂ΣΓ .
Again, by Theorem 3.1 the Bowditch boundary ∂(GΓ , P) is obtained from
the CAT(0) boundary ∂ΣΓ by identifying limit set of each peripheral left
coset of a subgroup in P to a point. Let g be this quotient map. The two
points w1 and w2 do not lie in limit sets of peripheral left cosets of subgroups
in P. Therefore, g(w1 ) 6= g(w2 ) and they are non-parabolic
 points in the
Bowditch boundary ∂(GΓ , P). Moreover, g ∂ΣΓ − {w1 , w2 } = ∂(GΓ , P) −
{g(w1 ), g(w2 )} and limit set of each peripheral left coset of a subgroup in P
lies completely inside ∂ΣΓ − {w1 , w2 }.
We observe that for any two points s1 , s2 ∈ ∂ΣΓ − {w1 , w2 } satisfying
g(s1 ) = g(s2 ) the two points s1 and s2 both lie in some limit set C of a
peripheral left coset of a subgroup in P. Also, each subgroup in P is oneended, so C is connected. Therefore, s1 and s2 lie in the same connected
component of ∂ΣΓ − {w1 , w2 }. This implies that if U and V are different
components of ∂ΣΓ −{w1 , w2 }, then g(U )∩g(V ) = ∅. Therefore, ∂(GΓ , P)−
{g(w1 ), g(w2 )} is not connected. This implies that {g(w1 ), g(w2 )} is a nonparabolic cut pair of the Bowditch boundary ∂(GΓ , P).

Theorem 3.9. Let Γ be a simplicial graph and J be a collection of induced
proper subgraphs of Γ. Assume that the right-angled Coxeter groups GΓ is
one-ended and hyperbolic relative to the collection P = { GJ | J ∈ J }, and
suppose each subgroup in P is one-ended. If the Bowditch boundary ∂(GΓ , P)
has a non-parabolic cut pair, then Γ has a separating complete subgraph
suspension. Moreover, if Γ has a separating complete subgraph suspension
whose non-adjacent vertices do not lie in the same subgraph J ∈ J, then the
Bowditch boundary ∂(GΓ , P) has a non-parabolic cut pair.
Proof. Since GΓ is one-ended, the Bowditch boundary ∂(GΓ , P) is connected.
If the Bowditch boundary ∂(GΓ , P) is a circle, then GΓ is virtually a surface
group, and the peripheral subgroups are the boundary subgroups of that
surface by Theorem 6B in [Tuk88]. This is a contradiction because each peripheral subgroup is one-ended. Therefore, the Bowditch boundary ∂(GΓ , P)
is not a circle. We now assume that the Bowditch boundary ∂(GΓ , P) has
a non-parabolic cut pair {u, v}. Then u is obviously a non-parabolic local
cut point. Therefore by Theorem 1.1 in [Haub], GΓ splits over a two-ended
subgroup H.
Since GΓ splits over a two-ended subgroup H, there is an induced subgraph K of Γ which separates Γ such that GK is contained in some conjugate
of H by Theorem 1 in [MT09]. Because the group GΓ is one-ended, the group

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

u

Γ1

17

v

Γ2

Figure 2. Graph Γ1 has no separating complete subgraph
suspension while graph Γ2 has cut pair (u, v) such that u and
v do not lie in the same 4-cycle.

GK is two-ended. This implies that K is a complete subgraph suspension.
The remaining conclusion is obtained from Proposition 3.8.

Example 3.10. Let Γ1 and Γ2 be the graphs in Figure 2. Then GΓ1 and
GΓ2 are both one-ended. Let J1 and J2 be the sets of all induced 4-cycles
of Γ1 and Γ2 , respectively. By Theorems 2.23 and 2.25, the collection Ji
is a peripheral structure of Γi for each i. Also, subgroups in each Pi =
{ GJ | J ∈ Ji } are virtually Z2 and thus one-ended. Moreover, for each i
no induced subgraph of a graph in Ji separates Γi . Thus by Theorem 3.5,
both Bowditch boundaries ∂(GΓ1 , P1 ) and ∂(GΓ2 , P2 ) have no cut points. So
in this case, we cannot use cut points to differentiate GΓ1 and GΓ2 up to
quasi-isometry. However, the graph Γ1 has no separating complete subgraph
suspension, so by Theorem 3.9 the Bowditch boundary ∂(GΓ1 , P1 ) has no
non-parabolic cut pair. Meanwhile, Γ2 has a cut pair (u, v) such that u
and v do not lie in the same subgraph in J2 . Again, by Theorem 3.9 the
Bowditch boundary ∂(GΓ2 , P2 ) has a non-parabolic cut pair. By Theorem
2.12, GΓ1 and GΓ2 are not quasi-isometric.
4. Relatively hyperbolic right-angled Coxeter groups with
Sierpinski carpet or sphere Bowditch boundary
In this section, we give a necessary and sufficient conditions for a relatively
hyperbolic right-angled Coxeter group whose defining graph has planar flag
complex to have the Sierpinski carpet or sphere as its Bowditch boundary.
We begin by recalling topological descriptions of the Sierpinski carpet and
Menger curve due to Whyburn [Why58] and Anderson [And58a, And58b],
respectively. Anderson’s result will be used in Section 5 to study nonhyperbolic right-angled Coxeter groups with Menger curve CAT(0) boundary.

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

18

Theorem 4.1 ([Why58], [And58a, And58b]). A compact space Σ is homeomorphic to the Sierpinski carpet if and only if it is 1–dimensional, planar,
connected, locally connected, and has no local cut points.
Let Σ be the Sierpinski carpet, then the non-separating embedded circles
(called peripheral circles) are pairwise disjoint, and form an infinite countable set.
A compact space Σ is homeomorphic to the Menger curve if and only if it
is 1–dimensional, connected, locally connected, has no local cut points, and
no non-empty open subset of Σ is planar.
We begin by introducing some terminology that we will use to characterize
a certain class of relatively hyperbolic right-angled Coxeter groups whose
boundaries are the Sierpinski carpet or 2-sphere. Some of these terms will
also be used in Section 5.
Definition 4.2. Let Γ be a finite simplicial graph. The flag complex of Γ
is the simplicial complex with 1-skeleton Γ and any complete subgraph of Γ
is the 1-skeleton of some simplex. Obviously, if Γ is triangle free, the flag
complex of Γ is Γ itself.
Definition 4.3. A graph is inseparable if it is connected, has no separating complete subgraph, no cut pair, and no separating complete subgraph
suspension. Obviously, a triangle free graph is inseparable if and only if it
is connected, has no separating vertex, no separating edge, no cut pair, and
no separating vertex suspension.
Definition 4.4. A graph K is an n–cycle extension if K is an n–cycle or K
is a join graph between an n–cycle and a complete graph. Note that if K is
an n–cycle extension subgraph of a triangle free graph, then K is an n–cycle.
An induced n–cycle extension subgraph K is strongly non-separating if no
induced subgraph of K separates the ambient graph.
Theorem 4.5 (Theorem A.2 in [DT]). Let K be a finite simplicial graph.
Then the right-angled Coxeter group GK is a hyperbolic virtual surface group
if and only if K is an n–cycle extension (n ≥ 5).
The following lemma is an easy exercise and is left to the reader.
Lemma 4.6. Let K be a finite simplicial graph. Then the right-angled
Coxeter group GK is a virtually Z2 group if and only if K is a 4–cycle
extension.
We first prove the sufficient conditions of the characterization (see Theorem 1.4). We recall Theorem 3.1 that the Bowditch boundary of a relatively
hyperbolic right-angled Coxeter group GΓ is obtained from the CAT(0)
boundary of the Davis complex ΣΓ by identifying the limit set of each peripheral left coset to a point. Therefore, we recall the following result due to
Świa̧tkowski [Ś] about CAT(0) boundaries of right-angled Coxeter groups.
We then investigate the limit set of each peripheral left coset in the CAT(0)
boundary and compute the quotient space ∂(GΓ , P).

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

19

Theorem 4.7 (Corollary 1.4 in [Ś]). Let Γ be a graph whose flag complex is
planar. Then the CAT(0) boundary ∂ΣΓ is the Sierpinski carpet if and only
if Γ is inseparable, distinct from a complete graph, and the flag complex of
Γ is distinct from a triangulation of S 2 .
Lemma 4.8. Let Γ be a graph whose flag complex is planar. Assume that
the graph Γ is inseparable and Γ has a non-trivial peripheral structure J that
consists of induced 4-cycle extension subgraphs. Let K be a strongly nonseparating induced 4-cycle extension or K a strongly non-separating induced
n–cycle extension (n ≥ 5) with the following properties:
(1) K does not contain any nonadjacent vertices of an induced 4-cycle
extension;
(2) No vertex outside K is adjacent to two non-adjacent vertices of K.
Then the limit set of the subgroup GK is a peripheral circle in the Sierpinski
carpet ∂ΣΓ .
Proof. Let J = J ∪ {K} and P = { GJ | J ∈ J }. By Theorem 2.23, the group
GΓ is relatively hyperbolic with respect to the collection P. We assume that
the limit set of the subgroup GK is a separating circle C of the Sierpinski
carpet ∂ΣΓ . By Theorem 3.1, the Bowditch boundary ∂(GΓ , P) is obtained
from the CAT(0) boundary ∂ΣΓ by identifying the limit set of each peripheral left coset of a subgroup in P to a point. Let f be this quotient map. Let
vGK be the point in ∂(GΓ , P) that is image of C under the map f . Since no
induced subgraph of K separates Γ, the point vGK is not a global cut point
of ∂(GΓ , P) by Theorem 3.5.
We observe that for any two points u, v ∈ ∂ΣΓ − C satisfying f (u) = f (v)
the two points u and v both lie in some circle C1 ⊂ ∂ΣΓ − C that is the
limit set of some peripheral subgroup. Therefore, u and v lie in the same
connected component of ∂ΣΓ − C. This implies that if U and V are different
components of ∂ΣΓ −C, then f (U )∩f (V ) = ∅. Therefore, vGK is the global
cut point which is a contradiction. Thus, the limit set of the subgroup GK
is a peripheral circle in the Sierpinski carpet ∂ΣΓ .

Proposition 4.9. Let Γ be a graph whose flag complex is planar. Assume
that Γ is inseparable and Γ has a non-trivial peripheral structure J that
consists of strongly non-separating induced 4-cycle extension graphs. Then
the group GΓ is relatively hyperbolic with respect to the collection P = { GJ |
J ∈ J } and the Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet.
Proof. By Theorem 4.7, the CAT(0) boundary ∂ΣΓ of the Davis complex ΣΓ
is the Sierpinski carpet. Also, by Lemma 4.8 the limit set of each peripheral
left coset is a peripheral circle of the Sierpinski carpet ∂ΣΓ . By Theorem
3.1, the Bowditch boundary ∂(GΓ , P) is obtained from the CAT(0) boundary
∂ΣΓ by identifying the limit set of each peripheral left coset to a point.
We now consider two cases. If all peripheral circles of the Sierpinski ∂ΣΓ
are limit sets of peripheral left cosets, then the Bowditch boundary ∂(GΓ , P)

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

20

is S2 . Otherwise, it is clear that the Bowditch boundary ∂(GΓ , P) is still the
Sierpinski carpet.

We now work on the necessary conditions for the characterization (see
Theorem 1.4).
Theorem 4.10 (Theorem 0.3 and Corollary 0.4 in [Dah05]). If (G, P) is
a relatively hyperbolic group whose boundary is the Sierpinski carpet or S2 ,
then each subgroup in P is virtually a surface group. Moreover, if (G, P) is a
minimal relatively hyperbolic structure, then each subgroup in P is virtually
Z2 .
Proposition 4.11. Let Γ be a graph whose flag complex is planar. Assume
that Γ has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }.
If the Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet, then Γ is
inseparable and each graph in J is a strongly non-separating 4-cycle extension
graph.
Proof. If the graph Γ is not connected or Γ has a separating complete
subgraph, then GΓ is not one-ended. Therefore, the Bowditch boundary
∂(GΓ , P) is not connected which is a contradiction. Thus, the graph Γ is
connected and has no separating complete subgraph. By Theorem 4.10,
each subgroup in P is virtually Z2 . Therefore by Lemma 4.6, each subgraph
in J must be a 4-cycle extension. Also, ∂(GΓ , P) has no global cut point.
Then each graph in J is strongly non-separating. Also, there is no cut pair
and no separating complete subgraph suspension of the ambient graph Γ
which is a subgraph of a graph in J. By way of contradiction, we assume
that graph Γ has a cut pair or a separating complete subgraph suspension.
Then, the non-adjacent vertices of the cut pair (or of the separating complete subgraph suspension) does not lie in the same J ∈ J. Therefore by
Proposition 3.8, the Bowditch boundary ∂(GΓ , P) has a non-parabolic cut
pair, a contradiction. Thus, Γ is inseparable.

In the following proposition, we further differentiate between the cases of
Sierpinski carpet and S2 Bowditch boundaries.
Proposition 4.12. Let Γ be a graph whose flag complex is planar. Assume
that Γ is inseparable graph and Γ has a non-trivial peripheral structure J that
consists of strongly non-separating induced 4-cycle extension graphs. Let
P = { GJ | J ∈ J }. Then the Bowditch boundary ∂(GΓ , P) is the Sierpinski
carpet if and only if Γ contains a strongly non-separating induced n–cycle
extension K (n ≥ 5) satisfying the following properties:
(1) K does not contain any nonadjacent vertices of an induced 4-cycle
extension;
(2) No vertex outside K is adjacent to two non-adjacent vertices of K.
S
Proof. Let S be the vertex set of Γ and P = P ∈P P . We first assume that
Γ contains a strongly non-separating induced n–cycle extension K (n ≥ 5)

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

21

with all the properties as above. We will prove that the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet. By Theorem 4.7 and Lemma 4.8, the
CAT(0) boundary ∂ΣΓ is the Sierpinski carpet and the limit set of the subgroup GK is a peripheral circle in ∂ΣΓ . This implies that not all peripheral
circles of the Sierpinski carpet ∂ΣΓ are limit sets of peripheral left cosets.
Therefore, the Bowditch boundary ∂(GΓ , P) is still the Sierpinski carpet as
in the proof of Proposition 4.9.
We now assume that the Bowditch boundary ∂(GΓ , P) is the Sierpinski
carpet. As in the proof of Proposition 4.9, there is some peripheral circle
C in ∂ΣΓ which is not the limit set of a peripheral left coset. Therefore,
this circle C still survives after identifying the limit set of each peripheral
left coset in ∂ΣΓ to a point to obtain the Bowditch boundary ∂(GΓ , P).
Moreover, the quotient map is GΓ –equivariant (see Theorem 3.1), so the
stabilizers of C via the actions of GΓ on ∂ΣΓ and ∂(GΓ , P) are the same.
Call this group Stab(C).
By the proof of Proposition 3.7 in [Dah05], Stab(C) acts as a uniform convergence group on C, which is clearly its limit set. Moreover, g Stab(C)g −1 ∩
Stab(C) is finite for each g ∈
/ Stab(C). Therefore by Theorem 9.9 of [Hru10],
Stab(C) is generated by a finite set T and the inclusion
Stab(C) ֒→ GΓ in
duces a quasi-isometric embedding Γ Stab(C), T ֒→ Γ(GΓ , S ∪P). By Theorem 1.5 in [Osi06], GΓ is hyperbolic relative to the collection P∪{Stab(C)}.
Therefore by Theorem 2.24, Stab(C) is the conjugate of some special subgroup GK of GΓ . Thus, the group GΓ is relatively hyperbolic with respect
to the collection P = P ∪ {GK }, and GK is the stabilizer of some translation
C1 of C.
Since the group GΓ is relatively hyperbolic with respect to the collection
P = P ∪ {GK }, the subgraph K satisfies Condition (1) and (2) of the proposition. Also, the limit set of GK is the peripheral circle C1 in ∂ΣΓ . Arguing
as in Proposition 4.9, we have that the Bowditch boundary ∂(GΓ , P) is either
S2 or the Sierpinski carpet. In particular, ∂(GΓ , P) has no global cut point.
Therefore, no induced subgraph of K separates Γ. Also, GK is virtually a
surface group by Theorem 4.10, and GK is not virtually Z2 . Therefore by
Theorem 4.5, K is a strongly non-separating n–cycle (n ≥ 5).

The following theorem is obtained from Propositions 4.9, 4.11, and 4.12.
With planar condition imposed on the flag complex of defining graphs, this
theorem gives a complete characterization of relatively hyperbolic rightangled Coxeter groups whose Bowditch boundaries are the Sierpinski carpet
or S2 .
Theorem 4.13. Let Γ be a graph whose flag complex is planar. Assume
that Γ has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }.
The Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet if and only
if Γ is inseparable and each graph in J is a strongly non-separating 4-cycle
extension graph.

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

Γ1

Γ2

22

Γ3

Figure 3. The three groups GΓ1 , GΓ2 , and GΓ3 all have
homeomorphic CAT(0) boundaries (Sierpinski carpet), but
they have different Bowditch boundaries with respect to their
minimal peripheral structures.
In addition, the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if
and only if Γ contains a strongly non-separating induced n–cycle extension
K (n ≥ 5) satisfying the following properties:
(1) K does not contain any nonadjacent vertices of an induced 4-cycle
extension;
(2) No vertex outside K is adjacent to two non-adjacent vertices of K.
We now restrict Theorem 4.13 to the case of 2–dimensional right-angled
Coxeter groups GΓ (i.e. Γ is triangle free and has at least one edge) to obtain a slightly simpler characterization of relatively hyperbolic right-angled
Coxeter groups whose boundaries are the Sierpinski carpet or S2 .
Corollary 4.14. Let Γ be a triangle free, planar graph. Assume that Γ
has a non-trivial peripheral structure J and let P = { GJ | J ∈ J }. The
Bowditch boundary ∂(GΓ , P) is S2 or the Sierpinski carpet if and only if Γ
is inseparable and each graph in J is a strongly non-separating 4-cycle.
In addition, the Bowditch boundary ∂(GΓ , P) is the Sierpinski carpet if
and only if Γ contains an induced strongly non-separating n–cycle K (n ≥ 5)
satisfying the following properties:
(1) K does not contain any nonadjacent vertices of an induced 4-cycle;
(2) No vertex outside K is adjacent to two non-adjacent vertices of K.
We now discuss some examples of Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups whose defining graphs are triangle free
and planar.
Example 4.15. Let Γ1 , Γ2 , and Γ3 be the graphs in Figure 3. Let J1 , J2 ,
and J3 be the sets of all induced 4-cycles of Γ1 , Γ2 , and Γ3 , respectively. By
Theorems 2.23 and 2.25, for each i the collection Ji is a peripheral structure
of Γi . Therefore, each group GΓi is hyperbolic relative to the collection
Pi = { GJ | J ∈ Ji }. Moreover, each Davis complex ΣΓi is a CAT(0) space
with isolated flats and its CAT(0) boundary is the Sierpinski carpet by
Theorem 4.7. However, their Bowditch boundaries are pairwise distinct.

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

23

In fact, the Bowditch boundary ∂(GΓ1 , P1 ) is S2 and the Bowditch boundary ∂(GΓ2 , P2 ) is the Sierpinski carpet by Theorem 4.13. In particular, this
implies that GΓ1 and GΓ2 are not quasi-isometric. Also, the graph Γ3 has
a separating induced 4-cycle. Then, the Bowditch boundary ∂(GΓ3 , P3 ) has
global cut points. Therefore, GΓ3 is not quasi-isometric to the other two
groups.
5. Non-hyperbolic right-angled Coxeter groups with Menger
curve boundary
In this section, we study the CAT(0) boundary of 2–dimensional rightangled Coxeter groups with isolated flats. In particular, we give conditions
which guarantee that the CAT(0) boundary of such a group will be the
Menger curve (see Proposition 5.3). We conclude the section with a pair of
examples of right-angled Coxeter groups with Menger Curve CAT(0) boundary and a show in Lemma 5.7 that these examples have non-homeomorphic
Bowditch boundaries.
Our work in this section is mainly based on the following results of the
first author in [Haua] and Mihalik-Tschantz [MT09].
Theorem 5.1 (Theorem 1.2 in [Haua]). Let Γ be a group acting geometrically on a CAT(0) space X with isolated flats. Assume ∂X is 1–dimensional.
If Γ does not split over a virtually cyclic subgroup then one of the following
holds:
(1) ∂X is a circle
(2) ∂X is a Sierpinski carpet
(3) ∂X is a Menger curve.
Theorem 5.2 ([MT09]). Let Γ be a triangle free, connected graph with no
separating vertex and no separating edge. The right-angled Coxeter group
GΓ splits over a two-ended subgroup H if and only if Γ has a cut pair {u, v}
or has a separating vertex suspension σ. Moreover, the special subgroup
generated by the cut pair {u, v} or by the separating vertex suspension σ is
contained in some conjugate of H.
We are now ready for the main proposition of this section.
Proposition 5.3. Let Γ be a triangle free, non-planar graph with a nontrivial peripheral structure J that consists of induced 4-cycles, and assume
the Γ is inseparable. Let ΣΓ be the Davis complex of the right-angled Coxeter
group defined by Γ. If the CAT(0) boundary ∂ΣΓ is not a Sierpinski carpet,
then ∂ΣΓ is a Menger curve.
Proof. We first observe that ΣΓ is a CAT(0) space with isolated flats and
the group GΓ is hyperbolic relative to the collection P = { GJ | J ∈ J }. The
Davis complex is one-ended and 2–dimensional, because Γ is triangle free,
connected with no separating vertex, and has no separating edge. Therefore,
∂ΣΓ is connected and 1–dimensional (see [GO07]). The graph Γ has no cut

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

24

Ω2
Ω1

Figure 4. The two groups GΩ1 and GΩ2 have homeomorphic CAT(0) boundaries (Menger curve), but they have different Bowditch boundaries with respect to their standard
peripheral structures.
pair and no separating vertex suspension so by Theorem 5.2 the group GΓ
does not split over a two-ended subgroup. Because Γ is inseparable and
cannot contain a separating clique, we also have that GΓ does not split over
a finite group. Therefore by Theorem 5.1, ∂ΣΓ must be a circle, a Sierpinski
carpet, or a Menger curve.
Since Γ is not a 4–cycle and Γ has non-trivial peripheral structure J, the
CAT(0) boundary ∂ΣΓ contains infinitely many disjoint circles. Thus, ∂ΣΓ
is not a circle. By hypothesis ∂ΣΓ is not a Sierpinski carpet, so ∂ΣΓ must
be a Menger curve.

We now provide two examples illustrating Proposition 5.3. These examples were inspired by an example from [DHW]. Dani-Haulmark-Walsh take
a triple of a 3–manifold glued along a common boundary component to construct the first example of a non-hyperbolic group whose CAT(0) boundary
is the Menger curve. We apply this idea to right-angled Coxeter groups,
thereby constructing the first examples of non-hyperbolic right-angled Coxeter groups whose CAT(0) boundaries are Menger curve.
Construction 5.4. Let K1 , K2 , and K3 be copies of the graph Γ1 in Figure
3. In each graph Ki we fix an induced 4-cycle Ci and glue all 4-cycles Ci to
a 4-cycle C to obtain the graph Ω1 (see Figure 4). Let J1 be the collection
of all induced 4-cycles of Ω1 . The graph Ω1 is triangle free and inseparable.
Using Theorems 2.23 and 2.25, it is not hard to check J1 is a peripheral
structure of Ω1 . Next we show that the CAT(0) boundary of ΣΩ1 cannot be
the Sierpinski carpet by showing the ∂ΣΩ1 contains a non-planar subspace.

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

25

Lemma 5.5. Let Ω1 be the graph constructed above. Then the CAT(0)
boundary ∂ΣΩ1 is the Menger curve.
Proof. For each graph Ki let ΣKi be the associated Davis complex in ΣΩ1
that contains the identity e. Then all spaces ΣKi share the Davis complex
ΣC . We remind the reader that C is the 4-cycle on which each 4-cycle Ci
of the graph Ki is glued, and point out that limit set of ΣC is a circle.
Therefore, all limit sets ∂ΣKi meet in the limit set ∂ΣC . By Theorem 4.7
and Lemma 4.8, each limit set ∂ΣKi is a Sierpinski carpet and ∂ΣC is a
peripheral circle of ∂ΣKi . Therefore, the union of all limit sets ∂ΣKi is nonplanar (see [DHW]). Since ∂ΣΩ1 contains a non-planar subspace, it is not
the Sierpinski carpet. Thus by Proposition 5.3, ∂ΣΩ1 is a Menger curve. 
We now construct a variation Ω2 of the graph Ω1 such that the CAT(0)
boundaries of the two right-angled Coxeter groups GΩ1 and GΩ2 are homeomorphic, but their Bowditch boundaries with respect to minimal peripheral
structures are not.
Construction 5.6. Let L1 , L2 , and L3 be copies of the graph Γ2 in Figure 3. In each graph Li we fix a 5-cycle Di and we glue all 5-cycles Di
to a 5-cycle D to obtain the graph Ω2 (see Figure 4). Let J2 be the collection of all induced 4-cycles of Ω2 . Again, it is not hard to check that
J2 is a peripheral structure of Ω2 and that the graph Ω2 satisfies all the
conditions in Proposition 5.3. We now prove that the CAT(0) boundaries
of the right-angled Coxeter groups GΩ1 and GΩ2 are the same (the Menger
curve), but their Bowditch boundaries ∂(GΩ1 , P1 ) and ∂(GΩ2 , P2 ) are not
homeomorphic, where Pi = { GJ | J ∈ Ji }.
Lemma 5.7. Let Ω1 and Ω2 be the graphs constructed above. Then, the
CAT(0) boundaries of the two right-angled Coxeter groups GΩ1 and GΩ2 are
the Menger curve, but their Bowditch boundaries ∂(GΩ1 , P1 ) and ∂(GΩ2 , P2 )
are not homeomorphic where each peripheral structure Pi of GΩi is constructed above. Moreover, the Bowditch boundary ∂(GΩ2 , P2 ) is not homeomorphic to S2 or the Sierpinski carpet.
Proof. For each graph Li let ΣLi be the associated Davis complex in ΣΩ2
that contains the identity e, and let ΣD be the associated Davis complex for
the common 5-cycle D. The proof that ∂ΣΩ2 is a Menger curve is similar to
the proof of Lemma 5.5. We note that by Lemma 4.8 the limit set ∂ΣD is a
peripheral circle in each Sierpinski carpet ∂ΣLi . However, the limit set ∂ΣD
is not the limit set of a peripheral left coset because D is not an induced
4–cycle of Ω2 .
We now claim that the Bowditch boundaries ∂(GΩ1 , P1 ) and ∂(GΩ2 , P2 )
are not homeomorphic, where Pi = { GJ | J ∈ Ji }. Indeed, the Bowditch
boundary ∂(GΩ1 , P1 ) has a global cut point since the induced 4-cycle C
separates the graph Ω1 . Meanwhile, there is no induced subgraph of an

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

26

induced 4–cycle in Ω2 that separates Ω2 . This implies the Bowditch boundary ∂(GΩ2 , P2 ) has no global cut point. Therefore, the Bowditch boundaries
∂(GΩ1 , P1 ) and ∂(GΩ2 , P2 ) are not homeomorphic.
We now prove that the Bowditch boundary ∂(GΩ2 , P2 ) is not homeomorphic to S2 or the Sierpinski carpet. By Theorem 3.1, the Bowditch boundary
∂(GΩ2 , P2 ) is obtained from the CAT(0) boundary ∂ΣΩ2 by identifying the
limit set of each peripheral left coset to a point, let f be the quotient map.
We will prove that f (∂ΣL1 ) ∪ f (∂ΣL2 ) ∪ f (∂ΣL3 ) is also a triple of the
Sierpinski carpets along a peripheral circle.
We first observe that for each 4-cycle J ∈ J2 the intersection J ∩ L1 is
a single edge or the whole 4-cycle J. Therefore, the intersection between
the limit set ∂ΣL1 and the limit set ∂(gGJ ) of a peripheral left coset gGJ is
empty or the whole limit set ∂(gGJ ). Moreover, the second case occurs if and
only if J is a 4-cycle in L1 and g is an element in GL1 . The image f (∂ΣL1 )
is obtained from ∂ΣL1 by identifying each peripheral circle on ∂ΣL1 which
is the limit set of a peripheral left coset gGJ , where g ∈ GL1 and J ⊂ L1 , to
a point. Therefore, the image f (∂ΣL1 ) is still a Sierpinski carpet. Similarly,
the images of f (∂ΣL2 ) and f (∂ΣL3 ) are also a Sierpinski carpets.
Lastly, all limit sets ∂ΣLi share the limit set ∂ΣD as a common subspace.
The limit set ∂ΣD is a peripheral circle in each Sierpinski carpet ∂ΣLi , but it
is not the limit set of a peripheral left coset. Therefore, the images f (∂ΣL1 ),
f (∂ΣL2 ), and f (∂ΣL3 ) pairwise intersect in the circle ∂ΣD . This implies
that the union f (∂ΣL1 ) ∪ f (∂ΣL2 ) ∪ f (∂ΣL3 ) is a triple of the Sierpinski
carpet along a peripheral circle. Since f (∂ΣL1 ) ∪ f (∂ΣL2 ) ∪ f (∂ΣL3 ) cannot
be topologically embedded into S2 or the Sierpinski carpet, the Bowditch
boundary ∂(GΩ2 , P2 ) is not homeomorphic to S2 or the Sierpinski carpet. 
Corollary 5.8. The groups GΩ1 and GΩ2 are not quasi-isometric.
Remark 5.9. In Corollary 4.14, with the additional assumption that the
defining graph is planar, we characterized 2-dimensional relatively hyperbolic right-angled Coxeter group GΓ whose Bowditch boundary ∂(GΓ , P) is
a Sierpinski or S2 . A natural question to ask is whether one can remove
“planar” as a hypothesis on the graph Γ in Corollary 4.14. However, we
have seen that for the graph Ω2 above the Bowditch boundary ∂(GΩ2 , P2 ) is
not either a Sierpinski carpet or an S2 . Thus we cannot disregard the planar
hypothesis in Corollary 4.14.
References
[And58a] R. D. Anderson. A characterization of the universal curve and a proof of its
homogeneity. Ann. of Math. (2), 67:313–324, 1958.
[And58b] R. D. Anderson. One-dimensional continuous curves and a homogeneity theorem. Ann. of Math. (2), 68:1–16, 1958.
[BD14] Jason Behrstock and Cornelia Druţu. Divergence, thick groups, and short conjugators. Illinois J. Math., 58(4):939–980, 2014.
[BDM09] Jason Behrstock, Cornelia Druţu, and Lee Mosher. Thick metric spaces, relative
hyperbolicity, and quasi-isometric rigidity. Math. Ann., 344(3):543–595, 2009.

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

[BH99]

27

Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature,
volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.
[BHS17] Jason Behrstock, Mark F. Hagen, and Alessandro Sisto. Thickness, relative hyperbolicity, and randomness in Coxeter groups. Algebr. Geom. Topol.,
17(2):705–740, 2017. With an appendix written jointly with Pierre-Emmanuel
Caprace.
[Bow01] B. H. Bowditch. Peripheral splittings of groups. Trans. Amer. Math. Soc.,
353(10):4057–4082, 2001.
[Bow12] B. H. Bowditch. Relatively hyperbolic groups. Internat. J. Algebra Comput.,
22(3):1250016, 66, 2012.
[Cap09] Pierre-Emmanuel Caprace. Buildings with isolated subspaces and relatively hyperbolic Coxeter groups. Innov. Incidence Geom., 10:15–31, 2009.
[Cap15] Pierre-Emmanuel Caprace. Erratum to “Buildings with isolated subspaces and
relatively hyperbolic Coxeter groups” [ MR2665193]. Innov. Incidence Geom.,
14:77–79, 2015.
[Dah05] François Dahmani. Parabolic groups acting on one-dimensional compact spaces.
Internat. J. Algebra Comput., 15(5-6):893–906, 2005.
[Dav08] Michael W. Davis. The geometry and topology of Coxeter groups, volume 32 of
London Mathematical Society Monographs Series. Princeton University Press,
Princeton, NJ, 2008.
[DGP11] François Dahmani, Vincent Guirardel, and Piotr Przytycki. Random groups do
not split. Math. Ann., 349(3):657–673, 2011.
[DHW] Pallavi Dani, Matthew Haulmark, and Genevieve Walsh. Planarity and Menger
curve boundaries. In preparation.
[DST]
Pallavi Dani, Emily Stark, and Anne Thomas. Commensurability for certain
right-angled Coxeter groups and geometric amalgams of free groups. Submitted.
arXiv:1610.06245.
[DT]
Pallavi Dani and Anne Thomas. Bowditch’s JSJ tree and the quasi-isometry
classification of certain Coxeter groups. Preprint. arXiv:1402.6224.
[DT15] Pallavi Dani and Anne Thomas. Divergence in right-angled Coxeter groups.
Trans. Amer. Math. Soc., 367(5):3549–3577, 2015.
[GM08] Daniel Groves and Jason Fox Manning. Dehn filling in relatively hyperbolic
groups. Israel J. Math., 168:317–429, 2008.
[GO07] Ross Geoghegan and Pedro Ontaneda. Boundaries of cocompact proper CAT(0)
spaces. Topology, 46(2):129–137, 2007.
[Gro87] M. Gromov. Hyperbolic groups. In Essays in group theory, volume 8 of Math.
Sci. Res. Inst. Publ., pages 75–263. Springer, New York, 1987.
[Gro13] Bradley W. Groff. Quasi-isometries, boundaries and JSJ-decompositions of relatively hyperbolic groups. J. Topol. Anal., 5(4):451–475, 2013.
[Haua]
Matthew Haulmark. Boundary classification and 2-ended splittings of groups
with isolated flats. Submitted. arXiv:1704.07937.
[Haub]
Matthew Haulmark. Local cut points and splittings of relatively hyperbolic
groups. Submitted. arXiv:1708.02855.
[HK05] G. Christopher Hruska and Bruce Kleiner. Hadamard spaces with isolated flats.
Geom. Topol., 9:1501–1538, 2005. With an appendix by the authors and Mohamad Hindawi.
[Hru10] G.C. Hruska. Relative hyperbolicity and relative quasiconvexity for countable
groups. Algebr. Geom. Topol., 10(3):1807–1856, 2010.
[KK00] Michael Kapovich and Bruce Kleiner. Hyperbolic groups with low-dimensional
boundary. Ann. Sci. École Norm. Sup. (4), 33(5):647–669, 2000.
[Lev]
Ivan Levcovitz. Divergence of CAT(0) cube complexes and Coxeter groups. Submitted. arXiv:1611.04378.

ON BOUNDARIES OF RIGHT-ANGLED COXETER GROUPS

28

[MT09]

Michael Mihalik and Steven Tschantz. Visual decompositions of Coxeter groups.
Groups Geom. Dyn., 3(1):173–198, 2009.
[Osi06]
Denis V. Osin. Elementary subgroups of relatively hyperbolic groups and
bounded generation. Internat. J. Algebra Comput., 16(1):99–118, 2006.
[Ś]
Jacek Świa̧tkowski. Right-angled Coxeter groups with n-dimensional Sierpiński
compacta as boundaries. Submitted. arXiv:1603.02152.
[Tra13] Hung Cong Tran. Relations between various boundaries of relatively hyperbolic
groups. Internat. J. Algebra Comput., 23(7):1551–1572, 2013.
[Tuk88] Pekka Tukia. Homeomorphic conjugates of Fuchsian groups. J. Reine Angew.
Math., 391:1–54, 1988.
[Why58] G. T. Whyburn. Topological characterization of the Sierpiński curve. Fund.
Math., 45:320–324, 1958.
Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN 37240 USA
E-mail address: m.haulmark@vanderbilt.edu
Department of Mathematical Sciences, University of Wisconsin-Milwaukee,
P.O. Box 413, Milwaukee, WI 53201, USA
E-mail address: nguyen36@uwm.edu
Department of Mathematics, The University of Georgia, 1023 D. W. Brooks
Drive, Athens, GA 30605, USA
E-mail address: hung.tran@uga.edu

