Let be the Davis complex for a Coxeter system . The automorphism group of is naturally a locally compact group, and a simple combinatorial condition due to Haglund–Paulin and White determines when is nondiscrete. The Coxeter group may be regarded as a uniform lattice in . We show that many such also admit a nonuniform lattice , and an infinite family of uniform lattices with covolumes converging to that of . It follows that the set of covolumes of lattices in is nondiscrete. We also show that the nonuniform lattice is not finitely generated. Examples of 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 .
Cite this article
Anne Thomas, Existence, covolumes and infinite generation of lattices for Davis complexes. Groups Geom. Dyn. 6 (2012), no. 4, pp. 765–801DOI 10.4171/GGD/174