Bowditch's JSJ tree and the quasi‐isometry classification of certain Coxeter groups

Abstract

Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian [Bowditch, Acta Math. 180 (1998) 145-186]. Our main result gives an explicit, computable 'visual' construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our construction we identify a large class of such groups for which the JSJ tree, and hence the visual boundary, is a complete quasi-isometry invariant, and thus the quasi-isometry problem is decidable. We also give a direct proof of the fact that among the Coxeter groups we consider, the cocompact Fuchsian groups form a rigid quasi-isometry class. In Appendix B, written jointly with Christopher Cashen, we show that the JSJ tree is not a complete quasi-isometry invariant for the entire class of Coxeter groups we consider.We prove Theorem 1.2 in Section 3, and an explicit statement of this theorem appears as Theorem 3.37. A consequence of our description is Corollary 1.3. For Γ satisfying Standing Assumptions 1.1, there exists an algorithm to compute the JSJ tree of the right-angled Coxeter group W Γ .Corollary 1.5. The quasi-isometry problem is decidable for the class of right-angled Coxeter groups W Γ with Γ ∈ G.As a consequence of Bowditch's construction of the JSJ tree and Theorem 1.4, we also obtain that the visual boundary is a complete invariant for the same class of groups.Corollary 1.6. Let Γ, Λ ∈ G (where G is as defined above the statement of Theorem 1.4). Then W Γ and W Λ are quasi-isometric if and only if ∂W Γ and ∂W Λ are homeomorphic.The visual boundaries of hyperbolic groups have been investigated byŚwiątkowski and coauthors in a series of papers [31,[39][40][41][42]. In particular,in [31] Martin andŚwiątkowski consider visual boundaries of fundamental groups of graphs of groups with finite edge groups. Suppose now that W Γ and W Λ are 2-dimensional hyperbolic right-angled Coxeter groups with infinitely many ends. When applied to the above-mentioned tree-of-groups decompositions for W Γ and W Λ , the results of [31] imply that ∂W Γ and ∂W Λ are homeomorphic if and only if the corresponding sets of homeomorphism types of visual boundaries of 1-ended vertex groups are the same. Our results thus have applications to determining the homeomorphism type of ∂W Γ when W Γ has infinitely many ends.The work of Tukia [44], Gabai [22] and Casson-Jungreis [15] shows that the cocompact Fuchsian groups form a rigid quasi-isometry class, that is, any finitely generated group which is quasi-isometric to a Fuchsian group is cocompact Fuchsian. Our last application gives a direct proof of this fact for 2-dimensional right-angled Coxeter groups.Theorem 1.7. Any 2-dimensional right-angled Coxeter group which is quasi-isometric to a cocompact Fuchsian group is cocompact Fuchsian.We give a precise statement as Theorem 4.2 and prove this result in Section 4.2.We now discuss several quasi-isometry invariants that have previously been considered for right-angled Coxeter groups. In...

Detecting a subclass of torsion-generated groups

“…The isomorphism type of the Bass-Serre tree of the JSJ decomposition of a group in C is a complete quasi-isometry invariant, as shown by Malone [Mal10] for a subclass of groups in C called geometric amalgams of free groups and by Cashen-Martin [CM17] in the general setting; see also related work of Dani-Thomas [DT17]. Furthermore, there is a one-to-one correspondence between isomorphism types of JSJ trees of groups in C and (equivalence classes of) certain finite matrices called degree refinements, which are algorithmically computed from the JSJ decomposition.…”

Quasi‐isometric groups with no common model geometry

“…Remark Groups in the class are quasi‐isometric to certain right‐angled Coxeter groups, including those with defining graph (and nerve) a planar graph called a 3‐convex generalized ‐ graph ; see [, Definition 1.6; ; ] for de finitions and background. If is such a right‐angled Coxeter group, then the JSJ decomposition of is similar to the JSJ decomposition of a group in as in Remark ; in particular, the JSJ decomposition of has underlying graph constructed for some .…”

Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups

“…JSJ decomposition of right-angled Coxeter groups. In this section we recall the results we will need from [7]. We also establish some technical lemmas needed for our constructions in Section 3, which use similar arguments to those in [7].…”