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

TL;DR

This paper constructs a family of uniform lattices acting on Davis complexes with a non-discrete set of covolumes, revealing new insights into the structure of these lattices and their covolume sets.

Contribution

It demonstrates that under certain conditions, the set of covolumes of uniform lattices is non-discrete and contains rationals with large prime divisors, providing explicit constructions.

Findings

The set of covolumes is non-discrete under certain assumptions.

Constructed lattices as fundamental groups of complexes of groups.

Provided a new proof for an analogous result in regular right-angled buildings.

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.