Density of commensurators for uniform lattices of right-angled buildings

Angela Kubena and Anne Thomas

Abstract

Let $G$ be the automorphism group of a regular right-angled building $X$. The "standard uniform lattice" ${\mathrm{\Gamma }}_0$ in $G$ is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the commensurator of ${\mathrm{\Gamma }}_0$ is dense in $G$. For this, we develop a technique of "unfoldings" of complexes of groups. We use unfoldings to construct a sequence of uniform lattices ${\mathrm{\Gamma }}_n$ in $G$, each commensurable to ${\mathrm{\Gamma }}_0$, and then apply the theory of group actions on complexes of groups to the sequence ${\mathrm{\Gamma }}_n$. As further applications of unfoldings, we determine exactly when the group $G$ is nondiscrete, and we prove that $G$ acts strongly transitively on $X$.