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 $\Sigma$. The polyhedral automorphism group $G$ of $\Sigma$ 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 $\Sigma$. We conclude with a new proof of an already known analogous result for regular right-angled buildings.