Existence, covolumes and infinite generation of lattices for Davis complexes

Let $\Sigma$ be the Davis complex for a Coxeter system (W,S). Theautomorphism group G of $\Sigma$ is naturally a locally compact group, and asimple combinatorial condition due to Haglund--Paulin determines when G isnondiscrete. The Coxeter group W may be regarded as a uniform lattice in G. Weshow that many such G also admit a nonuniform lattice $\Gamma$, and an infinitefamily of uniform lattices with covolumes converging to that of $\Gamma$. Itfollows that the set of covolumes of lattices in G is nondiscrete. We also showthat the nonuniform lattice $\Gamma$ is not finitely generated. Examples of$\Sigma$ to which our results apply include buildings and non-buildings, andmany complexes of dimension greater than 2. To prove these results, weintroduce a new tool, that of "group actions on complexes of groups", and usethis to construct our lattices as fundamental groups of complexes of groupswith universal cover $\Sigma$.