Density of commensurators for uniform lattices of right-angled buildings

TL;DR

This paper proves the density of the commensurator of a standard uniform lattice in the automorphism group of a right-angled building, using unfoldings of complexes of groups, and explores implications for group discreteness and transitivity.

Contribution

It introduces a novel technique of unfoldings of complexes of groups to analyze lattices in automorphism groups of right-angled buildings.

Findings

The commensurator of the standard uniform lattice is dense in the automorphism group.

The technique of unfoldings helps determine when the automorphism group is nondiscrete.

The automorphism group acts strongly transitively on the building.

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. 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 \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.