Existence, covolumes and infinite generation of lattices for Davis complexes

Anne Thomas

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 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 \(\Sigma\).