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" \(\Gamma_0\) 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 \(\Gamma_0\) 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 \(\Gamma_n\) in \(G\), each commensurable to \(\Gamma_0\), and then apply the theory of group actions on complexes of groups to the sequence \(\Gamma_n\). As further applications of unfoldings, we determine exactly when the group \(G\) is nondiscrete, and we prove that \(G\) acts strongly transitively on \(X\).