Totally Reflected Groups

Abstract

A group G is totally reflected if it has a generating set S such that each edge in the Cayley graph Gamma(G,S) is inverted by some color-preserving reflection on the graph. For example, we will show that Coxeter groups and right-angled Artin groups are totally reflected and that a finitely generated abelian group is totally reflected if and only if its first invariant factor is even. We show that direct and free products of totally reflected groups are totally reflected. More generally, we develop a group construction called a right-angled product which generalizes free and direct products, and we show that a right-angled product of totally reflected groups is itself totally reflected.

A group G is strongly totally reflected if there exists a color-preserving reflection group G_R acting on Gamma(G,S) such that each edge in the graph is inverted by some reflection in G_R. We state and prove sufficient conditions for a totally reflected group to be strongly totally reflected and use these results to prove from a graphical perspective that any right-angled Artin group is commensurable with a right-angled Coxeter group. In particular, we show that both the right-angled Artin group A(Delta)=<S> and its associated right-angled Coxeter group A_r are finite-index subgroups of the group of color-preserving graph automorphisms of Gamma(A(Delta),S).