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

TL;DR

This paper proves that certain right-angled Coxeter groups are uniquely determined by their large-scale geometric structure, establishing a rigidity result for groups built from joins of generalized thick polygons.

Contribution

It demonstrates quasi-isometric rigidity for a specific class of right-angled Coxeter groups and provides a method to construct commensurable groups within this class.

Findings

Quasi-isometric groups in this class are isomorphic.

Rigidity holds for joins of generalized thick m-gons with m ≥ 3.

Construction method for commensurable 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 $m\geq 3$. 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.