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Existence, covolumes and infinite generation of lattices for Davis complexes

By A. Thomas


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

Publisher: 'European Mathematical Society Publishing House'
Year: 2012
DOI identifier: 10.4171/GGD/174
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Provided by: Enlighten
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