Recognizing right-angled Coxeter groups using involutions

Charles Cunningham, Andy Eisenberg, Adam Piggott and Kim Ruane

Vol. 284 (2016), No. 1, 41–77

DOI: 10.2140/pjm.2016.284.41

Abstract

We consider the question of determining whether or not 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 apply this process to a number of examples. Our new results imply several known results as corollaries. In particular, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group, and we recover an existing result stating that if $\Gamma$ satisfies a particular graph condition (called no SILs), then ${Aut}^0\left(W_{\Gamma }\right)$ is a right-angled Coxeter group.

Keywords

Coxeter group, involutions, graph theory, automorphisms

Mathematical Subject Classification 2010

Primary: 20F55, 20F65

Secondary: 05C75

Milestones

Received: 20 July 2015

Revised: 9 December 2015

Accepted: 1 March 2016

Published: 10 July 2016