Recognizing Right-Angled Coxeter Groups Using Involutions

TL;DR

This paper introduces an invariant called the involution graph to identify right-angled Coxeter groups, providing a method to construct or disprove such groups and proving their rigidity properties.

Contribution

It characterizes involution graphs of right-angled Coxeter groups and offers a process to recognize or rule out these groups from a given presentation.

Findings

Involution graphs characterize right-angled Coxeter groups.

A process to construct or disprove right-angled Coxeter presentations.

Proof of rigidity of the defining graph for these groups.

Abstract

We consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for constructing candidate right-angled Coxeter presentations for a given group or proving that one cannot exist. We provide some first applications. In addition, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group. We also recover a result stating that if the defining graph contains no SILs, then Aut^0(W) is a right-angled Coxeter group.