Recognizing right-angled Coxeter groups using involutions

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 Γ satisfies a particular graph condition (called no SILs), then Aut0.(W Γ) is a right-angled Coxeter group.