Geodesic
Existence, covolumes and infinite generation of lattices for Davis complexes
Anne Thomas1 orcid
1The University of Sydney, Australia
Groups, geometry, and dynamics, Tome 6 (2012) no. 4, pp. 765-801

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DOI

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.
DOI : 10.4171/ggd/174
Classification : 22-XX, 57-XX, 00-XX
Mots-clés : Lattice, Davis complex, Coxeter group, building, complex of groups

Anne Thomas  1

1 The University of Sydney, Australia 
Anne Thomas. Existence, covolumes and  infinite generation of lattices  for Davis complexes. Groups, geometry, and dynamics, Tome 6 (2012) no. 4, pp. 765-801. doi: 10.4171/ggd/174
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     number = {4},
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     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/174/}
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