A family of uniform lattices acting on a Davis complex with a non-discrete set of covolumes

2015/11/30 by Damian Sercombe, Sercombe, Damian
Computer Science · Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GR #math.GT #msc:20F55 #msc:20F65 #msc:22D05 #msc:22F50

paper · pdf · doi:10.48550/arxiv.1511.09286

arxiv created 2015/11/30 · arxiv updated 2015/12/01

Abstract

Let (W,S) be a Coxeter system with Davis complex Σ. The polyhedral automorphism group G of Σ is a locally compact group under the compact-open topology. If G is a discrete group (as characterised by Haglund--Paulin), then the set \mathcal V_u(G) of uniform lattices in G is discrete. Whether the converse is true remains an open problem. Under certain assumptions on (W,S), we show that \mathcal V_u(G) is non-discrete and contains rationals (in lowest form) with denominators divisible by arbitrarily large powers of any prime less than a fixed integer. We explicitly construct our lattices as fundamental groups of complexes of groups with universal cover Σ. We conclude with a new proof of an already known analogous result for regular right-angled buildings.

Related

• Existence, covolumes and infinite generation of lattices for Davis complexes
• Automorphisms of geometric structures associated to Coxeter groups
• Topological rigidity fails for quotients of the Davis complex
• Covolumes of uniform lattices acting on polyhedral complexes
• On the non-discreteness of automorphism groups of Cayley graphs of Coxeter groups