On the relative hyperbolicity and manifold structure of certain right-angled Coxeter groups
Matthew Haulmark Hoang Thanh Nguyen Hung Cong Tran










shapes.gates.logic.US,trees,positioning,arrows

  







thmTheorem[section]
prop[thm]Proposition
lem[thm]Lemma
cor[thm]Corollary
conj[thm]Conjecture
*mainMain Theorem
claimClaim[thm]

definition
defn[thm]Definition
rem[thm]Remark
conv[thm]Convention
exmp[thm]Example
ques[thm]Question
prob[thm]Problem
cons[thm]Construction
assu[thm]Assumptions
*sassuStanding Assumptions




















































 

 














 





















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The relative hyperbolicity and manifold structure of certain RACGs]On the relative hyperbolicity and manifold structure of certain right-angled Coxeter groups


Department of Mathematics

1326 Stevenson Center

Vanderbilt University

Nashville, TN 37240 USA
m.haulmark@vanderbilt.edu


Department of Mathematical Sciences

University of Wisconsin-Milwaukee

P.O. Box 413

Milwaukee, WI 53201

USA
nguyen36@uwm.edu



Department of Mathematics

The University of Georgia

1023 D. W. Brooks Drive

Athens, GA 30605

USA
hung.tran@uga.edu





In this article, we study the manifold structure and the relatively hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups.




[2000]
20F67, 
20F65 
[
    Hung Cong Tran
    December 30, 2023
=====================





§ INTRODUCTION

For each finite simplicial graph Γ the associated right-angled Coxeter group G_Γ has generating set S equal to the vertices of Γ, relations s^2=1 for each s in S and relations st = ts whenever s and t are adjacent vertices. The graph Γ is the defining graph of right-angled Coxeter group G_Γ and its flag complex Δ=Δ(Γ) is the defining nerve of the group. Therefore, sometimes we also denote the right-angled Coxeter group G_Γ by G_Δ. In geometric group theory, groups acting on (0) cube complexes are fundamental objects and right-angled Coxeter groups provide a rich source of such groups. The coarse geometry of right-angled Coxeter groups has been studied by Caprace <cit.>, Dani-Thomas <cit.>, Dani-Stark-Thomas <cit.>, Behrstock-Hagen-Sisto <cit.>, Levcovitz <cit.> and others. In this paper, we will study the boundary of relatively hyperbolic right-angled Coxeter groups and the geometry of right-angled Coxeter groups with planar nerves.





























 














 
  
















































































In this paper, we study the geometry of right-angled Coxeter groups of planar defining nerves. We will focus our attention on one-ended non-hyperbolic groups. We will also exclude the virtually ^2 groups. Thus, we assume the nerves of our groups satisfy the following conditions.

The planar flag complex Δ⊂S^2:

    
  * is connected with no separating vertices and no separating edges (G_Δ is one-ended);
    
  * contains at least one induced 4-cycle (G_Δ is not hyperbolic);
    
  * is not a 4-cycle and not a cone of a 4-cycle (G_Δ is not virtually ^2).





 §.§ Manifold group structure and relatively hyperbolic structure

Davis-Okun <cit.> and Droms <cit.> proved that the one-ended right-angled Coxeter group G_Δ is virtually the fundamental group of a 3–manifold M if the defining nerve Δ is planar. So, we can learn a lot about the geometry of G_Δ if we know the manifold type of M. This is the main motivation for proving the following:



Let Δ⊂S^2 be a flag complex satisfying Standing Assumptions. Then the right-angled Coxeter group G_Δ is virtually the fundamental group of a 3-manifold M with empty or toroidal boundary if and only if Δ=S^2 or the boundary of each region in S^2-Δ is a 4-cycle. Moreover, in the case G_Δ is virtually the fundamental group of a 3-manifold M with empty or toroidal boundary, there are four mutually exclusive cases:

    
    
  * If Δ is a suspension of some n-cycle (n≥ 4) or some broken line (i.e a finite disjoint union of vertices and finite trees with vertex degrees 1 or 2), then M is a Seifert manifold;
    
  *  If the 1-skeleton of Δ is 𝒞ℱ𝒮 (see Definition <ref>) and does not satisfy (1), then M is a graph manifold; 
    
  * 
    If the 1-skeleton of Δ has no separating induced 4-cycle and is not 𝒞ℱ𝒮, then M is a hyperbolic manifold with boundary; 
    
  * 
   If the 1-skeleton of Δ contains a separating induced 4-cycle and is not 𝒞ℱ𝒮, then M is a mixed manifold.





    
    
    
    
    
    
    
    
    
    



 In Theorem <ref> we characterize the associated manifold M of G_Δ with empty or toroidal boundary using the work in <cit.> on the Euler characteristic of torsion free finite index subgroups of right-angled Coxeter groups. For further classification of such a manifold M we investigate the relatively hyperbolic structure of G_Δ. As a consequence, we obtain a key result on all possible divergence of G_Δ.  



Let Γ be a graph whose flag complex Δ is planar. There is a collection  of 𝒞ℱ𝒮 subgraphs of Γ such that the right-angled Coxeter group G_Γ is relatively hyperbolic with respect to the collection =G_JJ∈. In particular, if G_Γ is one-ended, then the divergence of G_Γ is linear, quadratic, or exponential.






For the proof of Theorem <ref> we carefully investigate the tree structure of the defining nerve Δ and then follow an almost identical strategy to the proof of Theorem 1.6 in <cit.>. The result concerning the divergence of the group follows directly from <cit.> and <cit.>. 


We also characterize right-angled Coxeter groups with planar nerves such that they have non-trivial minimal peripheral structures with the Sierpinski carpet as their Bowditch boundaries. This result will contribute to the study of the coarse geometry of our groups and we will discuss it in the next section. 



Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Then G_Δ has a non-trivial minimal peripheral structure with Bowditch boundary the Sierpinski carpet if and only if all following conditions hold:

    
  * The boundary of some region of S^2-Δ is an n-cycle with n≥ 5;
    
  * The 1-skeleton of Δ has no separating induced 4-cycle, no cut pair, and no separating induced path of length 2.






 §.§ Coarse geometry


Theorem <ref> divides right-angled Coxeter groups with nerves satisfying Standing Assumptions into two main types: the ones which are virtually the fundamental group of a 3–manifold with empty or toroidal boundary which we call type A, and the ones which are not virtually the fundamental group of such a 3–manifold which we will call type B. Behrstock and Neumann summarize the rigidity results of 3–manifolds with empty or toroidal boundary from many authors (see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>) in Theorem A of  <cit.>. Thus, a right-angled Coxeter group of type A and a right-angled Coxeter group of type B are never quasi-isometric by Theorem A of <cit.>.

Theorem <ref> further divides right-angled Coxeter groups of type A into 4 subtypes (see Figure <ref>). It is well-known that the fundamental groups of the associated manifolds in the four different subtypes are not quasi-isometric (see Theorem 5.4 in <cit.>), so 

two right-angled Coxeter groups of type A are not quasi-isometric if they are of different subtypes. Thus, we must study each subtype to understand the quasi-isometry classification of groups of type A.


 
  
   
    
     








The collection of groups of subtype A.1 contains exactly three quasi-isometry equivalence classes. The first quasi-isometry equivalence class of groups of type A.1 contains only one right-angled Coxeter group with nerve a suspension of a 4–cycle. This right-angled Coxeter group is virtually ^3. The second quasi-isometry equivalence class consists of right-angled Coxeter groups whose nerve is a suspension of some n–cycle (n≥ 5). Each right-angled Coxeter groups in this second class is virtually the fundamental group of a closed Seifert manifold. The last quasi-isometry equivalence class consists of right-angled Coxeter groups with nerve a suspension of some broken line. Each right-angled Coxeter group in this class is virtually the fundamental group of a Seifert manifold with non-empty boundary.

The quasi-isometry classification of groups of subtype A.2 is much more complicated. We give a complete quasi-isometry classification of groups of subtype A.2. 



Let Δ⊂S^2 and Δ'⊂S^2 be two planar flag complexes satisfying Standing Assumptions such that their 1–skeletons are 𝒞ℱ𝒮 and not join graphs. Let T_r and T_r' be the visual decomposition trees of Δ and Δ' respectively. Then two groups G_Δ and G_Δ' are quasi-isometric if and only if T_r and T_r' are bisimilar.


The above theorem strengthens Theorem 1.1 of <cit.> by removing the condition “triangle free” from the hypothesis of the theorem. We remark that the visual decomposition trees of the defining nerves in Theorems <ref> are colored trees whose vertices are colored black and white, and they are constructed in Sections <ref> and <ref>. We also refer the reader to Section <ref> for the notion of bisimilarity among two-colored graphs.

We also classify groups of subtype A.2 which are quasi-isometric to right-angled Artin groups. The following theorem is an extension of Theorem 1.2 of <cit.>.



Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions with the 1–skeleton a non-join 𝒞ℱ𝒮. Let T_r be a visual decomposition tree of Δ. Then the following are equivalent:


  * The right-angled Coxeter group G_Δ is quasi-isometric to a right-angled Artin group.

  * The right-angled Coxeter group G_Δ is quasi-isometric to the right-angled Artin group of a tree of diameter at least 3.

  * The right-angled Coxeter group G_Δ is quasi-isometric to the right-angled Artin group of a tree of diameter exactly 3.

  * All vertices of the tree T_r are black.




All right-angled Coxeter groups of subtype A.3 are virtually fundamental groups of hyperbolic 3–manifolds. So far, the authors do not know a complete quasi-isometric classification for the groups in this subtype. By Schwartz Rigidity Theorem <cit.> (see also, for example, Theorem 24.1 <cit.>) the fundamental groups of two hyperbolic 3–manifolds are quasi-isometric if and only if they are commensurable. Therefore, we turn the quasi-isometry classification problem for right-angled Coxeter groups of type A.3 into a commensurability problem.


Classify all right-angled Coxeter groups which are virtually the fundamental group of a hyperbolic 3–manifold up to commensurability. 


All right-angled Coxeter groups of subtype A.4 are virtually the fundamental groups of mixed 3–manifolds. Therefore, the last conclusion of Theorem <ref> potentially allows us to use the work of Kapovich-Leeb <cit.> to understand the geometry of right-angled Coxeter groups of subtype A.4. The complete quasi-isometry classification of fundamental groups of mixed manifolds remains an open question. As a consequence, the authors do not know the complete quasi-isometry classification of groups of subtype A.4.

We note that all right-angled Coxeter groups of type B are non-trivially relatively hyperbolic (see Proposition <ref>). So, we divide these groups into three different subtypes based on their Bowditch boundary with respect to the minimal peripheral structure and whether they split over 2–ended subgroups (see Figure <ref>).




	
Right-angled Coxeter groups of subtype B.1 all have minimal peripheral structure whose Bowditch boundary is a Sierpinski carpet (see Theorem <ref>). Meanwhile, the Bowditch boundary of right-angled Coxeter groups of subtypes B.2 and B.3 with respect to a minimal peripheral structure must have a cut point or a non-parabolic cut pair by Theorems <ref> and <ref>. Therefore, a right-angled Coxeter group of subtype B.1 is not quasi-isometric to one of subtype B.2 and subtype B.3. 

A right-angled Coxeter group of subtype B.2 always splits over a 2–ended subgroup while a right-angled Coxeter group of subtype B.3 does not. Therefore, they are never quasi-isometric by the work of Papasoglu <cit.>. (Papasoglu showed that among 1-ended finitely presented groups that are not commensurable to surface groups, having a splitting over a 2-ended subgroup is a quasi-isometry invariant.) We hope to use the work of Cashen-Martin <cit.> to understand the coarse geometry of right-angled Coxeter groups of subtype B.2. 

It seems hard to obtain the complete quasi-isometry classification of groups of type B. However, the quasi-isometry classification of the ones of subtype B.1 seems reasonable. 

Classify up to quasi-isometry all relatively hyperbolic right-angled Coxeter groups whose Bowditch boundary is a Sierpinski carpet with respect to some minimal peripheral structure.




 §.§ Outline of the paper


In Section <ref>, we review some concepts in geometric group theory: (0) spaces, δ–hyperbolic spaces, (0) spaces with isolated flats, 3–manifold groups, relatively hyperbolic groups, (0) boundaries, Gromov boundaries, Bowditch boundaries, and peripheral splitting of relatively hyperbolic groups. We also recall some necessary results related to these concepts. In Section <ref>, we give descriptions of cut points and non-parabolic cut pairs of Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups (see Theorems <ref> and <ref>). In Section <ref>, we study the coarse geometry of right-angled Coxeter groups with planar defining nerves. We first analyze the tree structure of planar flag complexes (see Subsection <ref>). Then we use this structure to study the relatively hyperbolic structure, group divergence, and manifold structure of right-angled Coxeter groups with planar nerves. We give the proofs of Theorem <ref>, Theorem <ref>, and Theorem <ref> in Subsection <ref>. We improve the tree structure of planar flag complexes with 1–skeletons 𝒞𝒮ℱ graphs and give a complete quasi-isometry classification of right-angled Coxeter groups which are virtually graph manifold groups in Subsection <ref>. The proof of Theorem <ref> is also given in this subsection.
	


 §.§ Acknowledgments

All three authors would like to thank Chris Hruska for suggestions and insights. The authors are also grateful for the insightful comments of Jason Behrstock, Ivan Levcovitz, and Mike Mihalik that have helped improve the exposition of this paper. The first author would like to thank Genevieve Walsh for helpful conversations concerning to this paper. Lastly, all three authors are particularly grateful for the referee's helpful comments and suggestions. 








§ PRELIMINARIES










In this section, we review some concepts in geometric group theory: (0) spaces, δ–hyperbolic spaces, (0) spaces with isolated flats, relatively hyperbolic groups, (0) boundaries, Gromov boundaries, Bowditch boundaries, and peripheral splitting of relatively hyperbolic groups. We also use the work of Behrstock-Druţu-Mosher <cit.> and Groff <cit.> to prove that the Bowditch boundary is a quasi-isometry invariant among relatively hyperbolic groups with minimal peripheral structures. We discuss the work of Caprace <cit.>, Behrstock-Hagen-Sisto <cit.>, Dani-Thomas <cit.>, <cit.>, and <cit.> on peripheral structures of relatively hyperbolic right-angled Coxeter groups and divergence of right-angled Coxeter groups. We also discuss the work of Gersten <cit.> and Kapovich–Leeb <cit.> on divergence of 3–manifold groups. We also mention the concept of colored graphs and the bisimilarity equivalence relation on such graphs.



 §.§ (0) spaces, δ–hyperbolic spaces, and relatively hyperbolic groups

We first discuss (0) spaces, δ–hyperbolic spaces, Gromov boundaries, and (0) boundaries. We refer the reader to the book <cit.> for more details.


We say that a geodesic triangle Δ in a geodesic space X satisfies the CAT(0) inequality if d(x,y) ≤ d(x̅,y̅) for all points x, y on the edges of Δ and the corresponding points x̅, y̅ on the edges of the comparison triangle Δ̅ in Euclidean space ^2.



A geodesic space X is said to be a CAT(0) space if every triangle in X satisfies the (0) inequality.

If X is a (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 α_x,ξ: [0,∞) → X with α_x,ξ(0)=x and α_x,ξ(∞)= ξ. The CAT(0) boundary has a natural topology with a basis of local neighborhoods (not necessarily open) of any point ξ∈∂ X given by the sets U(ξ;x, R, ϵ) = ξ'∈∂ Xd(α_x,ξ(R),α_x,ξ'(R)≤ϵ), where x ∈ X, ξ∈∂ X, R >0 and ϵ >0.



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 (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 (0) space (see for example <cit.> for details).



We now review relatively hyperbolic groups and related concepts.


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:


  * ℋ^(0)= V×{{0}∪}.

  * ℋ^(1) = {((t, n), (t, n + 1))}∪((t_1, n), (t_2, n))d_T(t_1,t_2)≤ 2^n. We call edges of the first set vertical and of the second horizontal.




In <cit.>, the combinatorial horoball is described as a 2-complex, but we only require the 1-skeleton for the horoball in this paper.



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).




Let G be a finitely generated group and  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∈. For each left coset gP of subgroup P∈ let ℋ(gP) be the horoball based at a copy of the subgraph T_gP with vertex set gP of the Cayley graph Γ(G,S). The cusped space X(G,,S) is the union of Γ(G,S) with ℋ(gP) for every left coset of P∈, identifying the subgraph T_gP with the depth 0 subset of ℋ(gP). We suppress mention of S and  when they are clear from the context.




Let G be a finitely generated group and  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∈. If the cusped space X(G,,S) is δ–hyperbolic then we say that G is hyperbolic relative to  or that (G,) is relatively hyperbolic. Collection  is a peripheral structure, each group P ∈ is a peripheral subgroup and its left cosets are peripheral left cosets. The peripheral structure  is minimal if for any other peripheral structure Q on G, each P∈ is conjugate into some Q∈Q.



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 <cit.>). As a consequence, the concept of relatively hyperbolic group does not depend on the choice of finite generating set.

We say that a finitely generated group is non-trivially relatively hyperbolic if it is relatively hyperbolic with respect to some collection of proper subgroups.






 §.§ The Bowditch boundary


We now discuss the Bowditch boundary of a relatively hyperbolic group and prove that it is a quasi-isometry invariant when the peripheral structure is minimal. We also recall peripheral splitting of relatively hyperbolic groups.



Let (G,) 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∈. The Bowditch boundary, denoted ∂(G,), is the Gromov boundary of the associated cusped space, X(G,,S).



There is a natural topological action of G on the Bowditch boundary ∂(G,) that satisfies certain properties (see <cit.>).

Bowditch has shown that the Bowditch boundary does not depend on the choice of finite generating set (see <cit.>). 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,,S) and X(G,,T) are G–equivariantly homeomorphic (see <cit.>).

For each peripheral left coset gP the limit set of the associated horoball ℋ(gP) consists of a single point in ∂(G,), called the parabolic point labelled by gP, denoted v_gP. The stabilizer of the point v_gP is the subgroup gPg^-1. We call each infinite subgroup of gPg^-1 a parabolic subgroup and subgroup gPg^-1 a maximal parabolic subgroup.



The homeomorphism type of the Bowditch boundary was already known to be a quasi-isometry invariant of a relatively hyperbolic group with minimal peripheral structure (see Groff <cit.>). However, we combine the work of Groff <cit.> and Behrstock-Druţu-Mosher <cit.> to elaborate on the homeomorphism between Bowditch boundaries induced by a quasi-isometry between two relatively hyperbolic groups with minimal peripheral structures.



Let (G_1,_1) and (G_2,_2) be finitely generated non-trivially relatively hyperbolic groups with minimal peripheral structures. If G_1 and G_2 are quasi-isometric, then there is a homeomorphism f from ∂(G_1,_1) to ∂(G_2,_2) that maps the set of parabolic points of ∂(G_1,_1) bijectively onto the set of parabolic points of ∂(G_2,_2). Moreover, if parabolic point v of ∂(G_1,_1) is labelled by some peripheral left coset g_1P_1 in G_1 and the parabolic point f(v) of ∂(G_2,_2) is labelled by some peripheral left coset g_2P_2 in G_2, then P_1 and P_2 are quasi-isometric.



Fix generating sets S_1 and S_2 as in Definition <ref> for G_1 and G_2 respectively, then there is a quasi-isometry q:Γ(G_1,S_1)→Γ(G_2,S_2). Since the peripheral structures are minimal, the map q takes a peripheral left coset g_1P_1 of G_1 to within a uniform bounded distance of the corresponding peripheral left coset g_2P_2 of G_2 by Theorem 4.1 in <cit.>. In particular, P_1 and P_2 are quasi-isometric. Using the proof of Theorems 6.3 in <cit.>, we can extend q to the quasi-isometry q̂:X(G_1,_1,S_1)→ X(G_2,_2,S_2) between cusped spaces such that q̂ restricts to a quasi-isometric embedding on each individual horoball of X(G_1,_1,S_1) and the image of the horoball lies in some neighborhood of a horoball of X(G_2,_2,S_2). Therefore, there is a homeomorphism f induced by q̂ from ∂(G_1,_1) to ∂(G_2,_2) that maps the set of parabolic points of ∂(G_1,_1) bijectively onto the set of parabolic points of ∂(G_2,_2). Moreover, if the parabolic point v of ∂(G_1,_1) is labelled by some peripheral left coset g_1P_1 in G_1 and the parabolic point f(v) of ∂(G_2,_2) is labelled by some peripheral left coset g_2P_2 in G_2, then by the above observation P_1 and P_2 are quasi-isometric.




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  of subgroups if each element of  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 relative to a collection . A peripheral splitting of (G,) is a representation of G as a finite bipartite graph of groups, where  consists of precisely of the (conjugacy classes of) vertex groups of one color. Obviously, any peripheral splitting of (G,) is relative to  and over subgroups of elements of . Peripheral splittings of (G,) are closely related to cut points in the Bowditch boundary ∂(G,) (<cit.>).


 


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. 




 §.§ (0) spaces with isolated flats


In this section, we discuss the work of Hruska-Kleiner <cit.> on (0) spaces with isolated flats.


A k–flat in a (0) space X is an isometrically embedded copy of Euclidean space ^k for some k ≥ 2. In particular, note that a geodesic line is not considered to be a flat.



Let X be a (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.


  * There is a constant D such that every flat F ⊂ X lies in a D–neighborhood of some F' ∈ℱ.

  * For each positive r < ∞ there is a constant ρ = ρ(r) < ∞ so that for any two distinct flats F, F' ∈ℱ we have (N_r(F) ∩ N_r(F′))<ρ.

We say X has isolated flats if it has isolated flats with respect to some G–invariant set of flats.




Suppose X has isolated flats with respect to ℱ. For each F ∈ℱ the stabilizer _G(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.



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.



Let G be a group acting geometrically on a (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 (0) space with isolated flats has a “well-defined” (0) boundary, often denoted by ∂ G, by the following theorem.


Let G act properly, cocompactly, and isometrically on (0) spaces X and Y. If X has isolated flats, then so does Y, and
there is a G–equivariant homeomorphism ∂ X →∂ Y.




 §.§ 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 <cit.> and Behrstock-Hagen-Sisto <cit.> on peripheral structures of relatively hyperbolic right-angled Coxeter groups.


Given a finite simplicial graph Γ, the associated right-angled Coxeter group G_Γ is generated by the set S of vertices of Γ and has relations s^2 = 1 for all s in S and st = ts whenever s and t are adjacent vertices. Graph Γ is the defining graph of a right-angled Coxeter group G_Γ and its flag complex Δ=Δ(Γ) is the defining nerve of the group. Therefore, sometimes we also denote the right-angled Coxeter group G_Γ by G_Δ where Δ is the flag complex of Γ. 

Let S_1 be a subset of S. The subgroup of G_Γ generated by S_1 is a right-angled Coxeter group G_Γ_1, where Γ_1 is the induced subgraph of Γ with vertex set S_1 (i.e. Γ_1 is the union of all edges of Γ with both endpoints in S_1). The subgroup G_Γ_1 is called a special subgroup of G_Γ.



The right-angled Coxeter group G_Γ is one-ended if and only if Γ is not equal to a complete graph, is connected and has no separating complete subgraphs (see Theorem 8.7.2 in <cit.>).



Given a finite simplicial graph Γ, the associated Davis complex Σ_Γ is a cube complex constructed as follows. For every k–clique, T ⊂Γ, the special subgroup G_T is isomorphic to the direct product of k copies of Z_2. Hence, the Cayley graph of G_T is isomorphic to the 1–skeleton of a k–cube. The Davis complex Σ_Γ has 1–skeleton the Cayley graph of G_Γ, where edges are given unit length. Additionally, for each k–clique, T ⊂Γ, and coset gG_T, we glue a unit k–cube to gG_T ⊂Σ_Γ. The Davis complex Σ_Γ is a (0) space and the group G_Γ acts properly and cocompactly on the Davis complex Σ_Γ (see <cit.>).




Let Γ be a simplicial graph and  be a collection of induced subgraphs of Γ. Then the right-angled Coxeter group G_Γ is hyperbolic relative to the collection =G_JJ∈ if and only if the following three conditions hold:


  * If σ is an induced 4-cycle of Γ, then σ is an induced 4-cycle of some J∈.

  * For all J_1, J_2 in  with J_1≠ J_2, the intersection J_1∩ J_2 is empty or J_1∩ J_2 is a complete subgraph of Γ.

  * If a vertex s commutes with two non-adjacent vertices of some J in , then s lies in J.










Let 𝒯 be the class consisting of the finite simplicial graphs Λ such that G_Λ is strongly algebraically thick. Then for any finite simplicial graph Γ either: Γ∈𝒯, or there exists a collection  of induced subgraphs of Γ such that ⊂𝒯 and G_Γ is hyperbolic relative to the collection =G_JJ ∈ and this peripheral structure is minimal.



In Theorem <ref> we use the notion of strong algebraic thickness which is introduced in <cit.> and is a sufficient condition for a group to be non-hyperbolic relative to any collection of proper subgroups. We refer the reader to <cit.> for more details. 






















 §.§ Divergence of right-angled Coxeter groups and 3–manifold groups
 Roughly speaking, divergence is a quasi-isometry invariant that measures the circumference of a ball of radius n as a function of n. We refer the reader to <cit.> for a precise definition. In this section, we state some theorems about divergence of certain right-angled Coxeter groups and 3-manifold groups which will be used later in this paper.




  §.§.§ Divergence of right-angled Coxeter groups



The divergence of a right-angled Coxeter group is either exponential (if the group is relatively hyperbolic) or bounded above by a polynomial (if the group is strongly algebraically thick).



Before continuing, we will take a brief detour to define a property of graphs that will be relevant to our study of right-angled Coxeter groups. Given a graph Γ, define Γ^4 as the graph whose vertices are induced 4–cycles of Γ. Two vertices in Γ^4 are adjacent if and only if the corresponding induced 4-cycles in Γ have two nonadjacent vertices in common. 



A graph Γ is 𝒞ℱ𝒮 if Γ is the join Ω*K where K is a (possibly empty) clique and Ω is a non-empty subgraph such that Ω^4 has a connected component T such that every vertex of Ω is contained in a 4–cycle that is a vertex of T. If Γ is 𝒞ℱ𝒮, then we will say that the right-angled Coxeter group G_Γ is 𝒞ℱ𝒮. 




Let Γ be a finite, simplicial, connected graph which has no separating complete subgraph. Let G_Γ be the associated right-angled Coxeter group.


  * The group G_Γ has linear divergence if and only if Γ is a join of two graphs of diameters at least 2.

  * The group G_Γ has quadratic divergence if and only if Γ is 𝒞ℱ𝒮 and is not a join of two graphs of diameters at least 2.





  §.§.§ Divergence of 3–manifold groups


Let M be a compact, connected, orientable 3–manifold with empty or toroidal boundary. The 3–manifold M is geometric if its interior admits a geometric structure in the sense of Thurston. These structures are the 3–sphere, Euclidean 3–space, hyperbolic 3-space, S^2×, ℍ^2×, SL(2,), Nil and Sol. We note that a geometric 3–manifold M is Seifert fibered if its geometry is neither Sol nor hyperbolic.
A non-geometric 3–manifold can be cut along tori called JSJ tori so that each component is a hyperbolic 3-manifold or a Seifert fibered space (<cit.>, <cit.>).  Each component of this JSJ decomposition is a piece. The manifold M is called a graph manifold if all the pieces are Seifert fibered, otherwise it is a mixed manifold.



Let M be a non-geometric manifold. Then M is a graph manifold if and only if the divergence of π_1(M) is quadratic, and M is a mixed manifold if and only if the divergence of π_1(M) is exponential.






Let M be a compact, orientable 3–manifold whose fundamental group has linear divergence. 
Then M has the geometry of Sol, or M is a Seifert manifold. However, if the universal cover M̃ of M is a fattened tree crossed with , then M must be a Seifert manifold.






 §.§ Two-colored graphs and their bisimilarity classes


In this section, we briefly discuss two-colored graphs and the bisimilar equivalence relation on them. These materials are introduced by Behrstock-Neumann <cit.> to study the quasi-isometry classification of graph manifolds. In this paper, we also use two-colored graphs and the bisimilar equivalence relation to study the quasi-isometry classification of right-angled Coxeter groups G_Γ whose defining graphs Γ are 𝒞ℱ𝒮 and satisfy Standing Assumptions.



A two-colored graph is a graph Γ with a “coloring” c:V(Γ) →{b,w}.
A weak covering of a two-colored graph Γ' is a graph homomorphism f:Γ→Γ' which respects colors and has the property that for each v∈ V(Γ) and for each edge e' ∈ E(Γ') at f(v), there exists an e∈ E(Γ) at v with f(e)=e'.




Two-colored graphs Γ_1 and Γ_2 are bisimilar, written Γ_1∼Γ_2 if Γ_1 and Γ_2 weakly cover some common two-colored graph.



The bisimilarity relation ∼ is an equivalence relation. Moreover, each equivalence class has a unique minimal element up to isomorphism.



































§ BOWDITCH BOUNDARIES OF RELATIVELY HYPERBOLIC RIGHT-ANGLED COXETER GROUPS



In this section, we give descriptions of cut points and non-parabolic cut pairs of Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups (see Theorems <ref> and <ref>). The Bowditch boundary of relatively hyperbolic groups with minimal peripheral structures is a quasi-isometry invariant (see Theorem <ref>). So, these results can be applied, in certain cases, to differentiate two relatively hyperbolic RACGs in terms of quasi-isometry equivalence.

In <cit.>, the third author investigates the connection between the Bowditch boundary of a relatively hyperbolic group (G,) and the boundary of a (0) space X on which G acts geometrically. For relatively hyperbolic right-angled Coxeter groups, the relevant result from <cit.> can be stated as follows:



Let Γ be a finite simplicial graph. Assume that the right-angled Coxeter group G_Γ is relatively hyperbolic with respect to a collection  of its subgroups. Then the Bowditch boundary ∂ (G_Γ,) is obtained from the (0) boundary ∂Σ_Γ by identifying the limit set of each peripheral left coset to a point. Moreover, this quotient map is G_Γ-equivariant.


We now introduce some definitions concerning defining graphs of right-angled Coxeter groups that we will use to “visualize” cut points and non-parabolic cut points in the Bowditch boundary.


Let Γ_1 and Γ_2 be two graphs, the join of Γ_1 and Γ_2, denoted Γ_1*Γ_2, is the graph obtained by connecting every vertex of Γ_1 to every vertex of Γ_2 by an edge. If Γ_2 consists of distinct vertices u and v, then the join Γ_1*{u,v} is the suspension of Γ_1.




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 Γ_1 of Γ is a complete subgraph suspension if Γ_1 is a suspension of a complete subgraph σ of Γ. If σ is a single vertex, then Γ_1 is a vertex suspension.

An induced subgraph Γ_1 of Γ is separating if Γ_1 separates Γ. Thus, we may 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 to visualize cut points in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group. 



Let Γ be a simplicial graph and  a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter group G_Γ is one-ended, hyperbolic relative to the collection =G_JJ∈, and suppose each subgroup in  is one-ended. Let J_0 be an element in  such that some induced subgraph of J_0 separates the graph Γ. Then we can write Γ=Γ_1∪Γ_2 such that the following conditions hold:


  * Γ_1, Γ_2 are both proper induced subgraphs of Γ;

  * Γ_1∩Γ_2 is an induced subgraph of J_0.

  * For each J in , J lies completely inside either Γ_1 or Γ_2

 



Since each subgroup in  is one-ended, each graph J in  is connected and J has no separating complete subgraph by Theorem 8.7.2 in <cit.>. Let L be an induced subgraph of J_0 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 J_0 is a proper subgraph of Γ, Γ'_1-J_0 ≠∅ or Γ'_2-J_0≠∅ (say Γ'_1-J_0≠∅).

Let Γ_1=Γ'_1, Γ_2=Γ'_2∪ J_0. Then, Γ_1 is an induced proper subgraph of Γ. We now prove that Γ_2 is also an induced proper subgraph. Choose a vertex w∈Γ_1-J_0=Γ'_1-J_0. Since Γ_1∩Γ_2=Γ'_1∩(Γ'_2∪ J_0)⊂ J_0, the vertex w does not belong to Γ_2. Therefore, Γ_2 is a proper subgraph of Γ. We now prove that Γ_2 is induced. Let e be an arbitrary edge with endpoints u and v in Γ_2. If e is an edge of Γ'_2, then e is the edge of Γ_2. Otherwise, e is an edge of Γ'_1=Γ_1, because Γ=Γ'_1∪Γ'_2. In particular, u and v are also the vertices of Γ_1. Again, Γ_1∩Γ_2 is a subgraph of J_0. Then u and v are vertices of J_0. Therefore, e is an edge of J_0, because J_0 is an induced subgraph. Thus, e is also an edge of Γ_2. Thus, Γ_2 is an induced subgraph. This implies that Γ_1∩Γ_2 is an induced subgraph. We already checked that Γ_1∩Γ_2 is a subgraph of J_0, and by construction Γ=Γ_1∪Γ_2.

Now, we prove that for each J in , J lies completely inside either Γ_1 or Γ_2. By the construction, J_0 is a subgraph of Γ_2. Therefore, we only need to check the case where J≠ J_0. It suffices to show that J lies completely inside either Γ'_1 or Γ'_2. By Theorem <ref>, for each J≠ J_0 in  the intersection J∩ J_0 is empty or it is a complete subgraph of Γ. Also the intersection J∩ L is an induced subgraph of J∩ J_0 if J∩ L≠∅. Therefore, J∩ L is empty or it is a complete subgraph of Γ. Recall that each graph in  is connected and has no separating complete subgraph by our assumption that the peripheral subgroups are 1–ended. Therefore, J-L=J-(J∩ L) is connected for each J≠ J_0 in . 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 Γ_1 or Γ_2.










The following theorem describes cut points in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group using the defining graph.



Let Γ be a simplicial graph and  be a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter group G_Γ is one-ended, G_Γ is hyperbolic relative to the collection =G_JJ∈, and suppose each subgroup in  is also one-ended. Then each parabolic point v_gG_J_0 is a global cut point if and only if some induced subgraph of J_0 separates the graph Γ.



We first assume that some induced subgraph of J_0 separates the graph Γ. Let Γ_1 and Γ_2 be graphs satisfying all conditions of Lemma <ref>. 





This implies that G_Γ=G_Γ_1*_G_KG_Γ_2, G_Γ_1≠ G_Γ, and G_Γ_2≠ G_Γ. Since J lies completely inside either Γ_1 or Γ_2 for each J∈, each peripheral subgroup in  must be a subgroup of G_Γ_1 or G_Γ_2. Therefore, G_Γ splits non-trivially relative to  over the parabolic subgroup G_K≤ G_J_0. 

By the claim following Theorem 1.2 of <cit.> the parabolic point v_G_J_0 labelled by G_J_0 is a global cut point of ∂(G,). Also, the group G_Γ acts as a group of homeomorphisms on ∂(G,) and gv_G_J_0=v_gG_J_0. Thus, each parabolic point v_gG_J_0 is also a global cut point.

Now, we assume that some parabolic point v_gG_J_0 is a global cut point. Again, the group G_Γ acts as a group of homeomorphisms on ∂(G,) and gv_G_J_0=v_gG_J_0. Therefore, v_G_J_0 is also a global cut point. Thus the maximal peripheral splitting 𝒢 of (G,) is non-trivial, and G_J_0 is a vertex stabilizer which is adjacent to a component vertex group in 𝒢 (see <cit.> for details concerning maximal peripheral splittings). So, by Theorem 3.3 of <cit.> G_Γ splits non-trivially over a subgroup H of G_J_0. Theorem 1 of <cit.> implies that there is some induced subgraph K of Γ which separates Γ such that G_K is contained in some conjugate of H. Therefore, G_K is also contained in some conjugate of the peripheral subgroup G_J_0. Moreover, G_K and G_J_0 are both special subgroups of G_Γ. Thus, K is an induced subgraph of J_0. (This is a standard fact, and we leave the details to the reader.)










Now, we 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 <ref> to the problem of quasi-isometry classification of right-angled Coxeter groups.



Let Γ_1, Γ_2, and Γ_3 be the graphs in Figure <ref>. Observe that all groups G_Γ_i are one-ended. We will prove that groups G_Γ_1, G_Γ_2, and G_Γ_3 are not pairwise quasi-isometric by investigating their minimal peripheral structures.

In Γ_1, let K_1^(1) and K_1^(2) be induced subgraphs generated by {a_1,a_2,a_3,a_4,a_5} and {a_6,a_7,a_8,a_9,a_10}, respectively. Observe that Γ_1 has only six induced 4-cycles which are not subgraphs of K_1^(1) and K_1^(2). Denote these cycles by L_1^(i) (i=1,2,⋯, 6). Let _1 be the set of all graphs L_1^(i) and K_1^(j). By Theorems <ref> and <ref>, _1 is the minimal peripheral structure of Γ_1. Moreover, no induced subgraph of a graph in _1 separates Γ_1. Therefore by Theorem <ref>, G_Γ_1 is hyperbolic relative to the collection _1=G_JJ∈_1 and the Bowditch boundary ∂ (G_Γ_1,_1) has no global cut point.

Similarly, let K_2^(1) and K_2^(2) be the induced subgraphs of Γ_2 generated by {b_1,b_2,b_3,b_4,b_5} and {b_6,b_7,b_8,b_9,b_10}, respectively. Observe that Γ_2 has only four induced 4-cycles, denoted L_2^(i) (i=1,2,⋯, 4), such that each of them is not a subgraph of K_2^(1) and K_2^(2). Let _2 be the set of all graphs L_2^(i) and K_2^(j). Then by Theorems <ref> and <ref>, _2 is the minimal peripheral structure of Γ_2. Moreover, K_2^(1) and K_2^(2) are the only graphs in _2 which contain induced subgraphs that separate Γ_2. Therefore, G_Γ_2 is hyperbolic relative to the collection _2=G_JJ∈_2, the Bowditch boundary ∂ (G_Γ_2,_2) has global cut points and each of them is labelled by some left coset of G_K_2^(1) or G_K_2^(2) by Theorem <ref>.

Finally, let K_3^(1) be an induced subgraph of Γ_3 generated by {c_6,c_7,c_8,c_9,c_10}. The graph Γ_3 has only three induced 4-cycles, denoted L_3^(i) (i=1,2,3), such that each of them is not a subgraph of K_3^(1). Assume that L_3^(1) is the induced 4-cycle generated by {c_1,c_2,c_5,c_4}. Let _3 be the set of all graphs L_3^(i) and K_3^(1). Again, by Theorem <ref> and <ref> we have that _3 is the minimal peripheral structure of Γ_3. Moreover, K_3^(1) and L_3^(1) are the only graphs in _3 which contain induced subgraphs that separate Γ_3. Therefore, G_Γ_3 is hyperbolic relative to the collection _3=G_JJ∈_3, the Bowditch boundary ∂ (G_Γ_3,_3) has global cut points and each of them is labelled by some left coset of G_K_3^(1) or G_L_3^(1) by Theorem <ref>.

Note that all the groups in _i are one-ended. The Bowditch boundary ∂ (G_Γ_1,_1) has no global cut point, but the Bowditch boundaries ∂ (G_Γ_2,_2) and ∂ (G_Γ_3,_3) do. So, G_Γ_1 cannot be quasi-isometric to G_Γ_2 or G_Γ_3. Additionally, the Bowditch boundary ∂ (G_Γ_3,_3) has global cut points labelled by a left coset of G_L_3^(1). Meanwhile, no global cut point of the Bowditch boundary ∂ (G_Γ_2,_2) is labelled by the left coset of a peripheral subgroup which is quasi-isometric to G_L_3^(1). Therefore, G_Γ_2 and G_Γ_3 are not quasi-isometric.


In the remainder of this section, we describe non-parabolic cut pairs in Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups in terms of their defining graphs.



Let Γ be a simplicial graph and  be a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter group G_Γ is one-ended, hyperbolic relative to the collection =G_JJ∈, and suppose each subgroup in  is also one-ended. If Γ has a separating complete subgraph suspension whose non-adjacent vertices do not lie in the same subgraph J∈, then the (0) boundary ∂Σ_Γ has a cut pair and the Bowditch boundary ∂(G_Γ,) has a non-parabolic cut pair.



Let K be a separating complete subgraph suspension of Γ whose non-adjacent vertices u and v do not both lie in the same subgraph 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∈, a contradiction.

Let K̅=σ*{u,v}. We can easily verify the following properties of K̅.


  * For each J∈ the intersection K̅∩ J is empty or a complete subgraph.

  * No vertex outside K̅ is adjacent to the unique pair of nonadjacent vertices {u,v} of K̅.

  * K is an induced subgraph of K̅.

Therefore, the collection =∪{K̅} satisfies all the conditions of Theorem <ref>, which implies that G_Γ is hyperbolic relative to the collection =G_JJ∈.

Using an argument similar to that of Lemma <ref>, we can write Γ=Γ_1∪Γ_2 such that the following conditions hold:


  * Γ_1, Γ_2 are both proper induced subgraphs of Γ;

  * Γ_1∩Γ_2 is an induced subgraph L of K̅.

  * For each J in , J lies completely inside either Γ_1 or Γ_2

Therefore, we can prove that the Bowditch boundary ∂ (G,) has a global cut point v_G_K̅ stabilized by the subgroup G_K̅ by using an argument similar to the one in Theorem <ref>.

By Theorem <ref>, the Bowditch boundary ∂ (G_Γ,) is obtained from the (0) boundary ∂Σ_Γ by identifying the limit set of each peripheral left coset of a subgroup in  to a point. Let f be this quotient map. Since G_K̅ is two-ended, its limit set consists of two points w_1 and w_2 in ∂Σ_Γ. Therefore, f(w_1)=f(w_2)=v_G_K̅ and f(∂Σ_Γ-{w_1,w_2})=∂(G_Γ,)-{v_G_K̅}. Since ∂(G_Γ,)-{v_G_K̅} is not connected, the space ∂Σ_Γ-{w_1,w_2} is also not connected. This implies that {w_1,w_2} is a cut pair of the (0) boundary ∂Σ_Γ.

Again, by Theorem <ref> the Bowditch boundary ∂ (G_Γ,) is obtained from the (0) boundary ∂Σ_Γ by identifying the limit set of each peripheral left coset of a subgroup in  to a point. Let h be this quotient map. The two points w_1 and w_2 do not lie in limit sets of peripheral left cosets of subgroups in . Therefore, h(w_1)≠ h(w_2) and they are non-parabolic points in the Bowditch boundary ∂ (G_Γ,). Moreover, h(∂Σ_Γ-{w_1,w_2})=∂(G_Γ,)-{h(w_1),h(w_2)} and the limit set of each peripheral left coset of a subgroup in  lies completely inside ∂Σ_Γ-{w_1,w_2}.

We observe that for any two points s_1, s_2 ∈∂Σ_Γ -{w_1,w_2} satisfying h(s_1)=h(s_2) the two points s_1 and s_2 both lie in some limit set C of a peripheral left coset of a subgroup in . Also, each subgroup in  is one-ended, so C is connected. Therefore, s_1 and s_2 lie in the same connected component of ∂Σ_Γ -{w_1,w_2}. This implies that if U and V are different components of ∂Σ_Γ -{w_1,w_2}, then h(U)∩ h(V)=∅. Therefore, ∂(G_Γ,)-{h(w_1),h(w_2)} is not connected. This implies that {h(w_1),h(w_2)} is a non-parabolic cut pair of the Bowditch boundary ∂(G_Γ,).






















The following theorem describes non-parabolic cut pairs in Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups in terms of their defining graphs.



Let Γ be a simplicial graph and  be a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter group G_Γ is one-ended, G_Γ is hyperbolic relative to the collection =G_JJ∈, and suppose each subgroup in  is one-ended. If the Bowditch boundary ∂ (G_Γ,) 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 ∈, then the Bowditch boundary ∂ (G_Γ,) has a non-parabolic cut pair.



Since G_Γ is one-ended, G_Γ does not split over a finite group by the characterization of one-endedness in Stallings's theorem. Therefore by Proposition 10.1 in <cit.> the Bowditch boundary ∂ (G_Γ,) is connected. If the Bowditch boundary ∂ (G_Γ,) is a circle, then G_Γ is virtually a surface group, and the peripheral subgroups are the boundary subgroups of that surface by Theorem 6B in <cit.>. This is a contradiction because each peripheral subgroup is one-ended. Therefore, the Bowditch boundary ∂ (G_Γ,) is not a circle. We now assume that the Bowditch boundary ∂ (G_Γ,) has a non-parabolic cut pair {u,v}. Then u is obviously a non-parabolic local cut point. Therefore by Theorem 1.1 in <cit.>, G_Γ splits over a two-ended subgroup H.



Since G_Γ splits over a two-ended subgroup H, there is an induced subgraph K of Γ which separates Γ such that G_K is contained in some conjugate of H by Theorem 1 in <cit.>. Because the group G_Γ is one-ended, the group G_K is two-ended. This implies that K is a complete subgraph suspension. The remaining conclusion is obtained from Proposition <ref>.


Now, we 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 <ref> to the problem of quasi-isometry classification of right-angled Coxeter groups.





Let Γ_1 and Γ_2 be the graphs in Figure <ref>. Then G_Γ_1 and G_Γ_2 are both one-ended. Let _1 and _2 be the sets of all induced 4-cycles of Γ_1 and Γ_2, respectively. By Theorems <ref> and <ref>, the collection _i is the minimal peripheral structure of Γ_i for each i. Also, subgroups in each _i=G_JJ∈_i are virtually ^2 and thus one-ended. Moreover, for each i no induced subgraph of a graph in _i separates Γ_i. Thus by Theorem <ref>, both Bowditch boundaries ∂ (G_Γ_1,_1) and ∂ (G_Γ_2,_2) have no cut points. So in this case, we cannot use cut points to differentiate G_Γ_1 and G_Γ_2 up to quasi-isometry. However, the graph Γ_1 has no separating complete subgraph suspension, so by Theorem <ref> the Bowditch boundary ∂ (G_Γ_1,_1) has no non-parabolic cut pair. Meanwhile, Γ_2 has a cut pair (u,v) such that u and v do not lie in the same subgraph in _2. Again, by Theorem <ref> the Bowditch boundary ∂ (G_Γ_2,_2) has a non-parabolic cut pair. By Theorem <ref>, G_Γ_1 and G_Γ_2 are not quasi-isometric.









































































§ GEOMETRIC STRUCTURE OF RIGHT-ANGLED COXETER GROUPS WITH PLANAR DEFINING NERVES



In this section, we study the coarse geometry of right-angled Coxeter groups with planar defining nerves. We first analyze the tree structure of planar flag complexes. Then we use this structure to study the relatively hyperbolic structure, group divergence, and manifold structure of right-angled Coxeter groups with planar nerves. Finally, we give a complete quasi-isometry classification of right-angled Coxeter groups which are virtually graph manifold groups.


A simplicial complex Δ is called flag if any complete subgraph of the 1-skeleton of Δ is the 1-skeleton of a simplex of Δ. Let Γ be a finite simplicial graph. The flag complex of Γ is the flag complex with 1-skeleton Γ. A simplicial subcomplex B of a simplicial complex Δ is called full if every simplex in Δ whose vertices all belong to B is itself in B.

The flag complex of Δ is planar if it can be embedded into the 2-dimensional sphere S^2. From now on every time we consider a flag complex it will be as a subspace of the 2-dimensional sphere S^2. 


The following lemma provides necessary and sufficient conditions for right-angled Coxeter groups of planar nerves to be one-ended.



Let Δ⊂S^2 be a non-simplex planar flag complex. 
Then G_Δ is one-ended if and only if Δ is connected has no separating vertex and no separating edge.



Assume that our flag complex Δ is not a simplex. It is known that the right-angled Coxeter group G_Δ is one-ended if and only if Δ is connected with no separating simplex. When Δ⊂S^2 is a connected flag complex and Δ has a separating simplex, we observe that Δ has a separating vertex or a separating edge. 


The following lemma provides necessary and sufficient conditions for when a one-ended right-angled Coxeter group with planar nerve splits over a two-ended subgroup.



Let Δ⊂S^2 be a non-simplex connected flag complex with no separating vertex and no separating edge. Then G_Δ splits over a two-ended subgroup if and only if Δ has a cut pair or a separating induced path of length 2.


By Lemma <ref>, the right-angled Coxeter group G_Δ is one-ended. Hence, G_Δ splits over a two-ended subgroup if and only if Δ contains a separating full subcomplex which is a cut pair or a suspension of a simplex. Since Δ⊂S^2 is a connected flag complex with no separating vertex and no separating edge, it is clear that if Δ has a separating full subcomplex which is a suspension of a simplex then Δ has a cut pair or a separating induced path of length 2.


We now restrict our attention to right-angled Coxeter groups which are one-ended, non-hyperbolic, and not virtually ^2. Therefore, we need the following assumptions on defining nerves of right-angled Coxeter groups.

The planar flag complex Δ⊂S^2:

    
  * is connected with no separating vertices and no separating edges (G_Δ is one-ended);
    
  * contains at least one induced 4-cycle (G_Δ is not hyperbolic);
    
  * is not a 4-cycle and not a cone of a 4-cycle (G_Δ is not virtually ^2).






 §.§ Decomposition of planar flag complexes




In <cit.>, the second and thirds authors describe a tree-like decomposition for triangle-free planar graphs. This decomposition has “nice” vertex graphs and is one of the key ideas of <cit.>. The techniques of <cit.> also apply to planar flag complexes. Starting with some terminology, we review the basics of this construction in the setting of planar flag complexes.



An induced 4–cycle σ of a flag complex Δ separates Δ if Δ-σ has at least two components.






Let Δ⊂S^2 be a planar flag complex. An induced 4–cycle σ strongly separates Δ if Δ has non-empty intersection with both components of S^2-σ.
The complex Δ is called prime if Δ satisfies the following conditions:

    
  * Δ is connected with no separating vertex and no separating edge;
    
  * Δ is not a 4–cycle but contains at least one induced 4-cycle;
    
  * Δ has no strongly separating induced 4-cycle (i.e. each induced 4-cycle bounds a region of S^2-Δ).




The suspension of a non-triangle graph (not necessarily connected) with 3 vertices is prime.



The notion of strongly separating in Definition <ref> depends on the choice of embedding map of the ambient flag complex into the sphere S^2. This notion is also based on the Jordan Curve Theorem that S^2-σ has two components.







A prime flag complex Δ is called special if it the suspension of a non-triangle graph (not necessarily connected) with 3 vertices. We remark that there are only three possible special prime flag complexes (see Figure <ref>). 





The following lemma will help us understand the structure of prime flag complexes and it can be compared to Lemma 3.7 in <cit.>.



Let Δ⊂S^2 be a prime flag complex. If Δ contains a full subcomplex B which is a special prime flag complex, then Δ is exactly the complex B. In particular, the 1-skeleton of Δ is 𝒞ℱ𝒮 if and only if Δ is special.



Assume by way of contradiction that Δ is not equal B. Then there is a vertex of v of Δ which is not a vertex of B. Therefore, v is a point in S^2-B. Since B is a special prime flag complex, B is one of the complexes in Figure <ref>. By working on each case of B in Figure <ref>, we always can find an induced 4-cycle σ in B such that v and the unique vertex u of B-σ lies in different components of S^2-σ. This implies that σ strongly separates Δ. Therefore, Δ is not prime which is a contradiction.

We now prove that the 1-skeleton of Δ is 𝒞ℱ𝒮 if and only if Δ is special. If Δ is special, we can see from Figure <ref> that the 1-skeleton of Δ is 𝒞ℱ𝒮. We now assume that the 1-skeleton of Δ is 𝒞ℱ𝒮. Then Δ contains a full subcomplex B which is a special prime flag complex. Therefore, Δ must be exactly the complex B as we proved above. Thus, Δ is special.


The following lemma will play an important role in understanding the relatively hyperbolic structure of right-angled Coxeter groups of planar defining nerves (see Theorem <ref>).



Let Δ⊂S^2 be a non-special prime flag complex. Let α_1 and α_2 be two distinct induced 4-cycles of Δ which have non-empty intersection. Then α_1∩α_2 is a vertex or an edge of Δ. 



Assume by way of contradiction that α_1∩α_2 contains two non-adjacent vertices, then the subcomplex of Δ induced by α_1∪α_2 contains a special full subcomplex B. In particular, Δ contains the special full subcomplex B. Also Δ is a prime flag complex. Therefore, by Lemma <ref>, Δ=B is special which is a contradiction. Therefore, α_1∩α_2 is a vertex or an edge of Δ. 


The following definition is an extension of the concept of strong visual decomposition of triangle free planar graphs (see Definition 3.4 in <cit.>) to planar flag complexes.


Let Δ⊂S^2 be a planar flag complex and σ a strongly separating 4-cycle of Δ. Then S^2-σ has two components U_1 and U_2 which both intersect Δ. For each i=1,2, let Δ_i be σ together with components of Δ-σ in U_i. Then, Δ=Δ_1∪Δ_2 and Δ_1∩Δ_2=σ. We call the pair (Δ_1,Δ_2) a strong visual decomposition of Δ with respect to the given embedding of Δ into S^2. If the embedding is clear from the context, we just say the pair (Δ_1,Δ_2) is a strong visual decomposition of Δ along σ.


The proof of the following proposition is analogous to the proof of Proposition 3.11 in <cit.>. We leave the details to the reader.




Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Then there is a finite tree T that encodes the structure of Δ as follows:


  * Each vertex v of T is associated to a full subcomplex Δ_v of Δ which is prime. Moreover, Δ_v≠Δ_v' if v≠ v' and ⋃_v∈ V(T)^Δ_v=Δ.

  * Each edge e of T is associated to an induced 4–cycle Δ_e of Δ. Moreover, Δ_e≠Δ_e' if e≠ e'.


  * Two vertices v_1 and v_2 of T are endpoints of the same edge e if and only if Δ_v_1∩Δ_v_2=Δ_e. Moreover, if V_1 and V_2 are vertex sets of two components of T removed the midpoint of e, then (⋃_v∈ V_1Δ_v,⋃_v∈ V_2Δ_v) is a strong visual decomposition of Δ along Δ_e.

Moreover, the 1-skeleton of Δ is 𝒞ℱ𝒮 if and only if the 1-skeleton of Δ_v is also 𝒞ℱ𝒮 for each vertex v of T (i.e. Δ_v is a special prime complex by Lemma <ref>).






 §.§ Relatively hyperbolic structure and manifold structure of RACGs with planar defining nerves



In this subsection, we are going to discuss the proof of Theorem <ref>, Theorem <ref>, and Theorem <ref>. Before proving Theorem <ref>, we discuss some key concepts in the following remark. 
















Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Then the divergence of the right-angled Coxeter group G_Δ is linear if and only if the 1–skeleton of Δ is a join of two graphs of diameters at least 2 (see Theorem <ref>). Since Δ⊂S^2 is a planar flag complex satisfying Standing Assumptions, the 1–skeleton of Δ is a join of two graphs of diameters at least 2 if and only if Δ is a suspension of a graph K where K is an n–cycle for n≥ 4 or K has at least 3 vertices and it is a finite disjoint union of vertices and finite trees with vertex degrees 1 or 2. If K has at least 3 vertices and it is a finite disjoint union of vertices and finite trees with vertex degrees 1 or 2, then we call K a broken line. 


We begin with a few words about the proof of Theorem <ref>. The proof of Theorem <ref> is completely analogous to that of Theorem 1.6 of <cit.>, and we leave the details to the reader. The key ideas behind the proof of Theorem 1.6 of <cit.> are the tree structure of graphs and the fact that the intersection of two induced 4–cycles in a prime graph which is not a suspension of three points is empty, a vertex, or an edge. In the case of planar flag complexes, the analogous facts are addressed by Proposition <ref> and Lemma <ref>. We will use Theorem <ref> to prove Theorem <ref>.





    
    
    
    
    
    
    
    
   
  
 




    
    
    
    
    
    
    
    
    
    




We first show that G_Δ is virtually the fundamental group of a 3-manifold M with empty or toroidal boundary if and only if the boundary of each region, if it exists, in S^2-Δ is a 4-cycle.

We are now going to prove the necessity. Suppose that the boundary of each region, if it exists, in S^2-Δ is a 4-cycle. Let Γ be the 1-skeleton of Δ. Then the boundary of each region in S^2-Γ is a 3-cycle or a 4-cycle. The group G_Δ acts by reflections on a simply-connected 3-manifold N with fundamental domain a ball whose boundary is the cell structure on S^2 that is dual to Γ. Since the boundary of each region in S^2-Γ is a 3-cycle or a 4-cycle, the stabilizer of each vertex is _2^3 or D_∞× D_∞. Let H be a torsion-free finite index subgroup of G_Δ. The quotient space N/H has the property that the link of each vertex associated to a region with 4-cycle boundary (if it exists) is a torus, and we can thus remove a finite neighborhood from such a vertex (if it exists) to obtain a desired manifold M whose boundary is empty or a union of tori.

We now prove sufficiency. Assume that G_Δ has a finite index subgroup H such that H is the fundamental group of a 3-manifold M with empty or toroidal boundary. Then the boundary of each region in S^2-Γ must be a 3-cycle or a 4-cycle, otherwise the Euler characteristic χ(H) is negative which is a contradiction (see Section 3.b in <cit.>). Therefore, the boundary of each region, if it exists, in S^2-Δ is a 4-cycle. 

Next, we consider all possible types of 3–manifold M via graph theoretic properties on Δ. 
The fact (<ref>) is clearly true since a right-angled Coxeter group induced by an n-cycle (n≥ 4) or a broken line is virtually a surface group. For the fact (<ref>) we first observe that the 1-skeleton of Δ is not a join of two graphs of diameter at least 2. 
Therefore, by Theorem <ref> the divergence of G_Δ (also the divergence of π_1(M)) is quadratic. By Theorem <ref> M must be a graph manifold. 

For the facts (<ref>) and (<ref>) we note that if the 1-skeleton of Δ is not 𝒞ℱ𝒮, then then the divergence of G_Δ (also the divergence of π_1(M)) is exponential by Theorem <ref>. Therefore, M must be a hyperbolic manifold with boundary or a mixed manifold. If, in addition, Δ contains at least a separating induced 4-cycle, then M contains at least a JSJ torus and it must be a mixed manifold which proves (<ref>). 


For (<ref>) we see that Δ is non-special prime flag complex. Therefore, in this case, G_Δ is hyperbolic relative to the collection of all right-angled Coxeter subgroups G_σ where σ is an induced 4-cycles of Δ. Due to the peripheral structure of G_Δ, the JSJ decomposition of M cannot have a Seifert piece. Moreover, the JSJ decomposition of M must consist of a single piece, otherwise π_1(M) would split over ^2. 



This implies that M is a hyperbolic manifold with boundary.   

We now study the relatively hyperbolic structure of right-angled Coxeter groups which are not virtually the fundamental group of a 3-manifold with empty or toroidal boundary. We prove that each such group is hyperbolic relative to a collection of its proper subgroups and we then study its Bowditch boundary with respect to the minimal peripheral structure. The following two lemmas will help us study the relatively hyperbolic structure of right-angled Coxeter groups which are not virtually the fundamental group of a 3-manifold with empty or toroidal boundary.



Let Δ⊂S^2 be a planar flag complex. Let (Δ_1,Δ_2) be a strong visual decomposition of Δ along some induced 4–cycle σ. If R is a region of S^2-Δ, then R is a region of either S^2-Δ_1 or S^2-Δ_2. 



Let U and V be two regions of S^2-σ such that Δ_1-σ⊂ U and Δ_2-σ⊂ V. Since R is a connected subset in S^2-σ, R lies inside either U or V (say U). We now prove that R is also a region of S^2-Δ_1. We first observe that S^2-Δ⊂S^2-Δ_1⊂ (S^2-Δ)∪ V. Therefore R lies in some region R_1 of S^2-Δ_1. Since R_1 is a connected subset of S^2-σ,  R_1 lies either in U or V, say U. Moreover, R_1∩ U contains the non-empty set R. Therefore R_1 must lie in U which implies that R_1∩ V=∅. Thus R_1 is a connected subset of S^2-Δ. This implies that R_1 must lie in some region of S^2-Δ and the region must be R. Therefore, R=R_1 is also a region of S^2-Δ_1.




Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Let T be a finite tree that encodes the structure of Δ as in Proposition <ref>. Then for each region R of S^2-Δ there is a vertex v of T such that R is also a region of S^2-Δ_v. 



The lemma can be proved by induction on the number of vertices of tree T by using Lemma <ref>.


In the rest of this subsection, we prove Theorem <ref>. The following proposition shows the existence of a non-trivial relatively hyperbolic structure of right-angled Coxeter groups which are not virtually the fundamental group of a 3-manifold with empty or toroidal boundary. 




Let Δ⊂S^2 be a connected flag complex with no separating vertex and no separating edge. Assume that the boundary of some region in S^2-Δ is an n-cycle for some n≥ 5. Then the 1-skeleton of Δ is not 𝒞ℱ𝒮. In particular, the divergence of G_Δ is exponential (i.e. G_Δ is non-trivially relatively hyperbolic). 



Since the boundary of some region in S^2-Δ is an n-cycle for some n≥ 5, it follows that Δ is not a simplex, a 4-cycle, or cone on a 4-cycle. If Δ does not contain an induced 4-cycle, then it is clear that the 1-skeleton of Δ is not 𝒞ℱ𝒮. Therefore, we assume that Δ contains at least one induced 4-cycle. In this case, we note that Δ satisfies Standing Assumption.

Let T be a tree that encodes the structure of Δ as in Proposition <ref>. By Lemma <ref> there is a vertex v of T such that the boundary of some region in S^2-Δ_v is an n-cycle for some n≥ 5. This implies that Δ_v is not a special prime complex. Therefore, by Proposition <ref> the 1-skeleton of Δ is not 𝒞ℱ𝒮. By Theorem <ref> the divergence of right-angled Coxeter group G_Δ is exponential.


We now study the Bowditch boundary of the relatively hyperbolic right-angled Coxeter groups in Proposition <ref>. We begin with the definition of Sierpinski carpet and recall a result due to Świa̧tkowski <cit.> which gives sufficient conditions for when the (0) boundary of a right-angled Coxeter group satisfying Standing Assumptions is a Sierpinski carpet.


Let D_1, D_2,⋯ be a sequence of open disks in a 2-dimensional sphere S^2 such that


  * D̅_i ∩D̅_j =∅ for i≠ j,

  * diam(D_i) → 0 with respect to the round metric on S^2, and

  * ⋃ D_i is dense. 

Then X= S^2-⋃ D_i is a Sierpinski carpet. The circles C=∂D̅_i ⊂ X are called peripheral circles.




Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Suppose also that the following two conditions hold:

    
  * For any two distinct regions of S^2-Δ, the intersection of their boundaries is empty, or a vertex, or an edge;
    
  * The boundary σ of each region of S^2-Δ satisfies the condition that for any two vertices of σ staying at distance 2 in Δ any induced path of length 2 in the 1-skeleton of Δ connecting these vertices is entirely contained in σ.

Then the (0) boundary ∂Σ_Δ is the Sierpinski carpet.










We now give a proof of Theorem <ref>. 


We assume that Δ satisfies both Conditions (1) and (2) of Theorem <ref> and we will prove that G_Δ has a non-trivial minimal peripheral structure with Bowditch boundary the Sierpinski carpet. We first show that the (0) boundary ∂Σ_Δ is the Sierpinski carpet by using Theorem <ref>. Then we will use Theorem <ref> to prove that G_Δ has a non-trivial minimal peripheral structure with Bowditch boundary the Sierpinski carpet. Since the 1-skeleton of Δ has no cut pair, Δ satisfies Condition (1) of Theorem <ref>. Also, the 1-skeleton of Δ has no separating induced path of length 2. 

Now, we show that Δ also satisfies Condition (2) of Theorem <ref>. By the way of contradiction that the boundary σ of some region of S^2-Δ does not satisfy Condition (2). Then there is an induced path α of length 2 in the 1–skeleton of Δ such that σ∩α consists of two distinct vertices u and v. We now claim that α separates the 1–skeleton of Δ which leads to a contradiction. We first assume that Δ=α∪σ. Then Δ-α=σ-{u,v} is disconnected. Therefore, α separates Δ which is also its 1–skeleton. We now assume that there is a point w in Δ-(α∪σ). We observe that σ is the union of two paths σ_1 and σ_2 such that σ_1∩σ_2={u,v}. Since σ bounds a region R in S^2-Δ, the path α lies completely outside the region R. Therefore, S^2-(α∪σ) contains exactly three regions and R is one of them. Since R is also a region in S^2-Δ, the point w can not lie in R. Therefore, w must lie in the region of S^2-(α∪σ) which is bounded by α∪σ_1 or α∪σ_2 (say α∪σ_1). Therefore, α separates w from a vertex of σ_2 in the 1–skeleton of Δ. This implies that α separates the 1–skeleton of Δ which leads to a contradiction. Thus, Δ satisfies Condition (2) of Theorem <ref>. Therefore, the (0) boundary ∂Σ_Δ is the Sierpinski carpet. 

Since the 1-skeleton of Δ has no separating induced 4-cycle, each induced 4-cycle of Δ must bound a region of S^2-Δ. Therefore, the intersection of two induced 4-cycles of Δ is empty, or a vertex, or an edge. Thus by Theorem <ref> the group G_Δ is relatively hyperbolic with respect to the collection  of all right-angled Coxeter groups induced by some induced 4-cycle of Δ. We now prove the Bowditch boundary ∂ (G_Γ,) is a Sierpinski carpet. 

We first claim that if σ is the boundary of a region in S^2-Δ, then the limit set of subgroup G_σ is a peripheral circle of the Sierpinski carpet ∂Σ_Γ. We see that the intersection of σ with an induced 4-cycle of Δ is empty, or a vertex, or an edge. Therefore by Theorem <ref> again G_Δ is relatively hyperbolic with respect to the collection =∪{G_σ}. By Theorem <ref>, the Bowditch boundary ∂ (G_Γ,) is obtained from the (0) boundary ∂Σ_Γ by identifying the limit set of each peripheral left coset of a subgroup in  to a point. Let f be this quotient map. Let v_G_K be the point in ∂ (G_Γ,) that is the image of the limit set of subgroup G_σ under the map f. Suppose by way of contradiction that the limit set of the subgroup G_σ is a separating circle of the Sierpinski carpet ∂Σ_Γ. Then the point v_G_K is a global cut point of ∂ (G_Γ,). We know that σ bounds a region of S^2-Δ while the intersection of the boundaries of two distinct regions of S^2-Δ is empty, or a vertex, or an edge. This implies that no induced subgraphs of σ separates the 1-skeleton of Δ. Therefore by Theorem <ref> point v_G_K is not a global cut point of ∂ (G_Γ,) which is a contradiction. Thus the limit set of subgroup G_σ is a peripheral circle of the Sierpinski carpet ∂Σ_Γ.

By Theorem <ref> again, the Bowditch boundary ∂ (G_Γ,) is obtained from the Sierpinski carpet ∂Σ_Γ by identifying each peripheral circle of ∂Σ_Γ which is the limit set of a peripheral left coset of a subgroup in  to a point. Let h be this quotient map. Then h is G_Γ–equivariant. By Condition (1) some boundary γ of a region of S^2-Δ is an n-cycle with n≥ 5. Therefore, the limit set of subgroup G_γ is a peripheral circle C of the Sierpinski carpet ∂Σ_Γ by the above argument. Since G_γ is not a group in , the peripheral circle C is not collapsed to a point via the map h. 

We now prove that the Bowditch boundary ∂ (G_Γ,) is a Sierpinski carpet. Let ℒ be the collection of all translates of the peripheral circle C by group elements in G_Δ. Then all peripheral circles in ℒ survive via the map h. Therefore, we can consider ℒ as a collection of pairwise disjoint circles in ∂ (G_Γ,). Moreover, ℒ is a G_Γ-invariant collection in ∂ (G_Γ,) since the quotient map h is G_Γ-equivariant. Also the action of G_Γ on ∂ (G_Γ,) is minimal (i.e. the orbit of each single point is dense in ∂ (G_Γ,)) by <cit.>. Therefore, the union of all circles in ℒ is dense in ∂ (G_Γ,). Fix metrics on (0) boundary ∂Σ_Γ and Bowditch boundary ∂ (G_Γ,). We consider ℒ as a sequence (C_n) of circles and we need to prove that (C_n)→ 0 with respect to the metric on ∂ (G_Γ,). 

Let ϵ be an arbitrary positive number and let {U_α}_α∈Λ be a cover of ∂ (G_Γ,) consisting of the open ball with diameter ϵ. Then {h^-1(U_α)}_α∈Λ is an open cover of ∂Σ_Γ. Since (C_n)→ 0 with respect to the metric on ∂Σ_Γ and ∂Σ_Γ is a compact space, then each set C_n lies in some member of the cover {h^-1(U_α)}_α∈Λ for each n sufficiently large. Therefore, each set C_n=h(C_n) also lies in some member of the cover {U_α}_α∈Λ for each n sufficiently large. This implies that the diameter of each such circle C_n is less than ϵ with respect to the metric on ∂ (G_Γ,). This implies that (C_n)→ 0 with respect to the metric on ∂ (G_Γ,). Therefore, the Bowditch boundary ∂ (G_Γ,) is a Sierpinski carpet by <cit.>.

We now assume that G_Δ has a non-trivial minimal peripheral structure with Bowditch boundary the Sierpinski carpet. We will prove that Δ satisfies both Conditions (1) and (2). In fact, if Δ does not satisfy Condition (1), then G_Γ is virtually a 3-manifold group with empty boundary or tori boundary by Theorem <ref>. 
Therefore, the Bowditch boundary of G_Δ with respect to a non-trivial minimal peripheral structure is not the Sierpinski carpet which is a contradiction. Therefore, Δ must satisfy Condition (1). Also Δ must satisfy Condition (2). Otherwise by Theorem <ref> and Theorem <ref> the Bowditch boundary of G_Δ with respect to a non-trivial minimal peripheral structure contains a global cut point or a cut pair which is a contradiction.














 §.§ Quasi-isometry classification of virtually graph manifold group RACGs



In this subsection, we prove Theorem <ref>. This theorem generalizes Theorem 1.1 in <cit.> by removing the condition “triangle-free” from the hypotheses.

In the rest of this section, we will assume that the planar flag complex Δ⊂S^2 satisfies Standing Assumptions and the 1-skeleton of Δ is a non-join 𝒞ℱ𝒮 graph. In particular, Δ is not the suspension of an n-cycle. Moreover, Δ also does not contain a proper full subcomplex which is the suspension of an n-cycle since Δ⊂S^2 is a planar flag complex. Let T be a tree that encodes the structure of Δ as in Proposition <ref>. Since the 1–skeleton of Δ is 𝒞ℱ𝒮, it is shown in Proposition <ref> that each vertex subcomplex Δ_v is the suspension of a non-triangle graph with 3 vertices. For the purpose of obtaining the quasi-isometry classification of our 𝒞ℱ𝒮 right-angled Coxeter groups, the tree structure T in Proposition <ref> is not the correct tree structure to use. So, we now modify the tree T to obtain a new two-colored new tree that encodes the structure of Δ.




Step 1: We color an edge of T by one of two colors: red and blue as follows. Let e be an edge of T with vertices v_1 and v_2. If either Δ_v_1 or Δ_v_2 is a suspension of a path of length 2 we color the edge e red. Next, we consider two cases. If Δ_v_1 and Δ_v_2 have the same suspension points, then we color the edge e red. Otherwise, we color e blue.


Step 2: Let ℛ be the union of all red edges of T. We remark that ℛ is not necessarily connected. We form a new tree T_r from the tree T by collapsing each component C of ℛ to a vertex labelled by v_C and we associate each such new vertex v_C to the complex Δ_v_C=⋃_v∈ V(C)Δ_v. For each vertex v of T_r which is also a vertex of T we still assign v the complex Δ_v as in the previous tree T structure. We observe that for each vertex v of the new tree T_r the associated vertex complex Δ_v is either one of the first two complexes in Figure <ref> or a union of at least two special complexes which all share a pair of suspension vertices. Therefore, Δ_v is the suspension of a broken line ℓ_v which is distinct from a path of length 2. We call the number of induced 4-cycles of Δ_v⊂S^2 that bound a region in S^2-Δ_v the weight of v denoted by w(v). 
By construction, each edge e of the new tree T_r is also an edge of the old tree T. Therefore, we still assign the edge e in the new tree T_r the complex Δ_v as in the previous tree T structure. 

We observe that for each edge e of T_r with vertices v_1 and v_2 the induced 4-cycle Δ_e bounds a region in both S^2-Δ_v_1 and S^2-Δ_v_2. Therefore, the weight of each vertex v is always greater than or equal to the degree of v in T_r. Moreover, if v_1 and v_2 are two adjacent vertices in T_r, then suspension vertices of Δ_v_1 are vertices of ℓ_v_2 and similarly suspension vertices of Δ_v_2 are vertices of ℓ_v_1.




Step 3: We now color vertices of T_r. For each vertex v of T_r, the complex Δ_v is a suspension of a broken line set ℓ_v. Observe that in Step 2 that the weight of each vertex v is always greater than or equal to the degree of v in T. Therefore, we now color v black if its weight is strictly greater than its degree. Otherwise, we color v white.


We now summarize some key properties of the tree T_r.




  * Each vertex v of T_r is associated to a full subcomplex Δ_v of Δ that is a suspension of a broken line ℓ_v which is distinct from a path of length 2. 
The weight w(v) of each vertex v is required to be greater than or equal its degree in T_r. We color v black if its weight is strictly greater than its degree. Otherwise, we color v white.

  * Δ_v≠Δ_v' if v≠ v' and ⋃_v∈ V(T_r)^Δ_v=Δ.

  * Each edge e of T_r is associated to an induced 4–cycle Δ_e of Δ. If e≠ e', then Δ_e≠Δ_e'.

  * Two vertices v_1 and v_2 of T_r are endpoints of the same edge e if and only if Δ_v_1∩Δ_v_2=Δ_e. Moreover, the induced 4-cycle Δ_e bounds a region in both S^2-Δ_v_1 and S^2-Δ_v_2. Also, suspension vertices of Δ_v_1 are vertices of ℓ_v_2. Similarly, suspension vertices of Δ_v_2 are vertices of ℓ_v_1. Lastly, if V_1 and V_2 are vertex sets of two components of T_r removed the midpoint of e, then (⋃_v∈ V_1Δ_v)∩(⋃_v∈ V_2Δ_v)=Γ_e. 
 



Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions with the 1-skeleton a non-join 𝒞ℱ𝒮 graph. A tree that encodes the structure of Δ carrying Properties (1), (2), (3), and (4) as above is called a visual decomposition tree of Δ.



The existence of a visual decomposition tree for a planar flag complex satisfying Standing Assumptions with the 1-skeleton a non-join 𝒞ℱ𝒮 graph is guaranteed by Construction <ref>. 










Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions with the 1-skeleton a non-join 𝒞ℱ𝒮 graph. Let T_r be a visual decomposition tree of Δ. Since Δ is planar, G_Δ is virtually a 3–manifold group by Theorem <ref>. However, for the purpose of obtaining a quasi-isometry classification we will construct explicitly a 3–manifold Y on which G_Δ acts properly and cocompactly. We note that the construction of the manifold Y is associated to the graph T_r. Therefore, we can import the work of Behrstock-Neumann <cit.> to prove Theorem <ref>.



We now construct a 3-manifold Y on which the right-angled Coxeter group G_Δ acts properly and cocompactly. For each vertex v of T_r, the complex Δ_v is a suspension of a broken line ℓ_v of Δ. We now let b and c be suspension vertices and let Σ_v be the Davis complex associated to the broken line ℓ_v. We fatten the 1-skeleton of Σ_v to obtain a universal cover of a hyperbolic surface with boundary as follows: 


Let n be the number vertices of  ℓ_v. We replace each vertex of the 1-skeleton Σ^(1)_v by a regular n–gon with sides labelled by vertices of ℓ_v. We assume that two edges of such an n–gon labeled by a_1 and a_2 in V(ℓ_v) are adjacent if and only if either a_1 and a_2 are vertices of an edge of ℓ_v or the set {a_1, a_2,b,c} forms an induced 4–cycle of Δ_v that bounds a region in S^2-Δ_v. We also assume the length side of the n–gon is 1/2. We replace each edge E labelled by a_i by a strip E×[-1/4,1/4]. We label each side of length 1 of the strip E×[-1/4,1/4] by a_i and we identify the edge E to E×{0} of the strip. If u is an endpoint of the edge E of Σ^(1)_v, then the edge {u}×[-1/4,1/4] is identified to the side labelled by a_i of the n–gon that replaces u. 

We observe that for each pair of adjacent vertices a_1 and a_2 of ℓ_v we have an induced 4–cycle in Σ^(1)_v with one pair of opposite sides labeled by a_1 and the other pair of opposite sides labeled by a_2. We note that this 4–cycle bounds a 2–cell in Σ_v. In the “fattening” of Σ^(1)_v constructed above we also have such 4–cycles and we also fill each with a 2–cell as in Σ_v. We denote F(Σ_v) to be the resulting space (See Figure <ref>). Clearly, G_ℓ_v acts properly and cocompactly on the space F(Σ_v). Additionally, F(Σ_v) is a simply connected surface with boundary and each boundary component of F(Σ_v) is an infinite concatenation of edges labelled by a_1 and a_2 where a_1 and a_2 are vertices of ℓ_v such that the set {a_1, a_2,b,c} forms an induced 4–cycle that bounds a region in S^2-Δ_v. We denote such a boundary by α_a_1,a_2.

The group G_{b,c} acts on the line α that is a concatenation of edges labelled by b and c by edge reflections. Let P_v=F(Σ_v)×α and equip P_v with the product metric. Then, the group G_Δ_v acts properly and cocompactly on P_v in the obvious way. The space P_v is the universal cover of the trivial circle bundle of a hyperbolic surface with nonempty boundary.

Moreover, for each pair of vertices a_1 and a_2 of ℓ_v such that the set {a_1, a_2,b,c} forms an induced 4–cycle that bounds a region in S^2-Δ_v the right-angled Coxeter groups generated by {a_1,a_2,b,c} acts on the boundary α_a_1,a_2×α in a way analogous to its action on the Davis complex. We label this plane by {a_1,a_2,b,c}. 

If v_1 and v_2 are two adjacent vertices in T_r, then the pair of suspension vertices (a_1,a_2) of Δ_v_1 is a pair of vertices of ℓ_v_2 and the pair of suspension vertices (b_1,b_2) of Δ_v_2 is a pair of of vertices of ℓ_v_1. Moreover, the set {a_1, a_2,b_1,b_2} forms an induced 4–cycle that bounds a region in both S^2-Δ_v_1 and S^2-Δ_v_2. Therefore, the spaces P_v_1 and P_v_2 have two Euclidean planes which are labeled by {a_1,a_2,b_1,b_2} as constructed above. Thus, using the Bass-Serre tree T̃_r of the decomposition of G_Δ as tree T_r of subgroups we can form a 3-manifold Y by gluing copies of P_v appropriately and we obtain a proper, cocompact action of G_Δ on Y.


We are ready to prove the quasi-isometry classification theorem. The proof is identical with the proof of (3) in Theorem 1.1 <cit.>.




We can color vertices of the Bass-Serre tree T̃_r so that the quotient map q: T̃_r→ T_r preserves the vertex coloring. Moreover, the map q: T̃_r→ T_r is a week covering. Therefore, the trees T̃_r and T_r are bisimilar. (See Definitions <ref> and <ref> for the definitions of weak covering and bisimilarity.) We observe that a vertex of T̃_r is colored black if and only if the corresponding Seifert manifold contains a component of the boundary of Y. Using the proof of Theorem 3.2 in <cit.>, we obtain the proof of Theorem <ref>.







Let Δ and Δ' be flag complexes in Figure <ref>. A visual decomposition tree T_r of Δ is shown in the same figure with the following information. The complex Δ_u_1 is the suspension of the broken line with three vertices a_1, a_3, and a_5 with two suspension vertices a_6 and a_7. The complex Δ_u_2 is the suspension of the broken line with three vertices a_2, a_6, and a_7 with two suspension vertices a_1 and a_3. The complex Δ_u_3 is the suspension of the broken line with three vertices a_4, a_6, and a_7 with two suspension vertices a_3 and a_5. The complex Δ_u_4 is the suspension of the broken line with three vertices a_6, a_7, and a_8 with two suspension vertices a_1 and a_5. We note, that the vertices u_2, u_3, and u_4 all have weight 2 and the vertex u_1 has weight 3. By comparing their degrees, the vertices u_2, u_3, and u_4 are colored black while u_1 is colored white.

Similarly, a visual decomposition tree T_r' of Δ' is also shown in Figure <ref> with the following information. The complex Δ_v_1 is the suspension of the broken line with four vertices b_1, b_3, b_5, and b_9 with two suspension vertices b_6 and b_7. The complex Δ_v_2 is the suspension of the broken line with three vertices b_2, b_6, and b_7 with two suspension vertices b_1 and b_3. The complex Δ_v_3 is the suspension of the broken line with three vertices b_4, b_6, and b_7 with two suspension vertices b_3 and b_5. The complex Δ_v_4 is the suspension of the broken line with three vertices b_6, b_7, and b_8 with two suspension vertices b_1 and b_9. We observe that the vertices v_2, v_3, and v_4 all have weight 2 and the vertex v_1 has weight 4. By comparing their degrees we see that all vertices of T_r' are colored black. Thus, the visual decomposition trees T_r and T_r' are not bisimilar even though they are isomorphic if we ignore the vertex colors. Therefore, the groups G_Γ and G_Γ' are not quasi-isometric.
 
































































































































































































































































































































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