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

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 Vu(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 Vu(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.