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 Gamm
a_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.
Let $Sigma$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $Sigma$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin determines when G is nondiscrete. The Coxeter group W m
ay be regarded as a uniform lattice in G. We show that many such G also admit a nonuniform lattice $Gamma$, and an infinite family of uniform lattices with covolumes converging to that of $Gamma$. It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice $Gamma$ is not finitely generated. Examples of $Sigma$ to which our results apply include buildings and non-buildings, and many complexes of dimension greater than 2. To prove these results, we introduce a new tool, that of group actions on complexes of groups, and use this to construct our lattices as fundamental groups of complexes of groups with universal cover $Sigma$.
The goal of this paper is to present a number of problems about automorphism groups of nonpositively curved polyhedral complexes and their lattices, meant to highlight possible directions for future research.
