Quasi-isometric rigidity of a class of right-angled Coxeter groups

Abstract

We establish quasi-isometric rigidity for a class of right-angled Coxeter groups. Let ${\Gamma }_1,{\Gamma }_2$ be joins of finite generalized thick $m$-gons with $m\ge 3$. We show that the corresponding right-angled Coxeter groups are quasi-isometric if and only if ${\Gamma }_1,{\Gamma }_2$ are isomorphic. We also give a construction of commensurable right-angled Coxeter groups.