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" 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 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 n in G, each commensurable to 0, and then apply the theory of group actions on complexes of groups to the sequence 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.
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