Existence, covolumes and infinite generation of lattices for Davis complexes

Thomas, A. (2012) Existence, covolumes and infinite generation of lattices for Davis complexes. Groups, Geometry and Dynamics, 6(4), pp. 765-801. (doi: 10.4171/GGD/174)

Abstract

Let Σ be the Davis complex for a Coxeter system (W,S). The automorphism group G of Σ 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 Γ, and an infinite family of uniform lattices with covolumes converging to that of Γ. It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice Γ is not finitely generated. Examples of Σ 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 Σ.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Thomas, Dr Anne
• Authors: Thomas, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Groups, Geometry and Dynamics
• ISSN: 1661-7207
• ISSN (Online): 1661-7215