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.
Publication Source (Journal or Book title)
Journal of Topology
Dani, P., & Thomas, A. (2017). Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups. Journal of Topology, 10 (4), 1066-1106. https://doi.org/10.1112/topo.12033