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" Γ_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 Γ_0 is dense in G. This result was also obtained by Haglund. For our proof, we develop carefully a technique of "unfoldings" of complexes of groups. We use unfoldings to construct a sequence of uniform lattices Γ_n in G, each commensurable to Γ_0, and then apply the theory of group actions on complexes of groups to the sequence Γ_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.

29 pages, 10 figures. Version 2: revised abstract and introduction, to appear in J. Group Theory