Existence, covolumes and infinite generation of lattices for Davis complexes

Anne Thomas

Abstract

Let $\mathrm{\Sigma }$ be the Davis complex for a Coxeter system $(W,S)$. The automorphism group $G$ of $\mathrm{\Sigma }$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund-Paulin determines when $G$ is nondiscrete. The Coxeter group $W$ may be regarded as a uniform lattice in $G$. We show that many such $G$ also admit a nonuniform lattice $\mathrm{\Gamma }$, and an infinite family of uniform lattices with covolumes converging to that of $\mathrm{\Gamma }$. It follows that the set of covolumes of lattices in $G$ is nondiscrete. We also show that the nonuniform lattice $\mathrm{\Gamma }$ is not finitely generated. Examples of $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$.