JournalsggdVol. 6, No. 4pp. 765–801

Existence, covolumes and infinite generation of lattices for Davis complexes

  • Anne Thomas

    The University of Sydney, Australia
Existence, covolumes and  infinite generation of lattices  for Davis complexes cover
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Abstract

Let Σ\Sigma be the Davis complex for a Coxeter system (W,S)(W,S). The automorphism group GG of Σ\Sigma is naturally a locally compact group, and a simple combinatorial condition due to Haglund–Paulin and White determines when GG is nondiscrete. The Coxeter group WW may be regarded as a uniform lattice in GG. We show that many such GG also admit a nonuniform lattice Γ\Gamma, and an infinite family of uniform lattices with covolumes converging to that of Γ\Gamma. It follows that the set of covolumes of lattices in GG is nondiscrete. We also show that the nonuniform lattice Γ\Gamma is not finitely generated. Examples of Σ\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 Σ\Sigma.

Cite this article

Anne Thomas, Existence, covolumes and infinite generation of lattices for Davis complexes. Groups Geom. Dyn. 6 (2012), no. 4, pp. 765–801

DOI 10.4171/GGD/174