Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen

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. 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 an appendix, 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.