Existence, covolumes and infinite generation of lattices for Davis complexes

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.