Density of commensurators for uniform lattices of right-angled buildings

Abstract.

Let G be the automorphism group of a regular right-angled building X. The “standard uniform lattice” is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the commensurator of is dense in G. This result was also obtained by Haglund (2008). For our proof, we develop carefully a technique of “unfoldings” of complexes of groups. We use unfoldings to construct a sequence of uniform lattices , each commensurable to , and then apply the theory of group actions on complexes of groups to the sequence . As further applications of unfoldings, we determine exactly when the group G is nondiscrete, and prove that G acts strongly transitively on X.