The relative hyperbolicity and manifold structure of certain right-angled Coxeter groups

Abstract

In this article, we first study the Bowditch boundary of relatively hyperbolic right-angled Coxeter groups. More precisely, we give "visual descriptions" of cut points and non-parabolic cut pairs in the Bowditch boundary of relatively hyperbolic right-angled Coxeter groups. Then we study the manifold structure and the relatively hyperbolic structure of right-angled Coxeter groups with planar nerves. We use these structures to study the quasi-isometry problem for this class of right-angled Coxeter groups.

Complete topological descriptions of certain Morse boundaries

“…Tran conjectured that the Morse boundary of a right-angled Coxeter group is totally disconnected if and only if every induced cycle of length greater than four in the defining graph contains a pair of non-adjacent vertices that are contained in an induced 4-cycle [28]. As Tran pointed out to us, the one-skeleton of a 3-cube satisfies this condition and its associated right-angled Coxeter group is virtually a finite volume hyperbolic 3-manifold with cusps [14]. It follows from our theorem that this is a counterexample to Tran's conjecture.…”