Existence, covolumes and infinite generation of lattices for Davis complexes

Abstract

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 and White determines when $G$ is nondiscrete. The Coxeter group $W$ may 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 $\Sigma$.