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

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.