Existence, covolumes and infinite generation of lattices for Davis complexes

TL;DR

This paper investigates the properties of lattices in the automorphism groups of Davis complexes, showing the existence of nonuniform lattices, infinite families of uniform lattices with converging covolumes, and the non-finite generation of certain lattices.

Contribution

It introduces a new method using complexes of groups to construct and analyze lattices in Davis complexes, revealing non-discreteness of covolume sets and non-finite generation of some lattices.

Findings

The set of covolumes of lattices in G is nondiscrete.

Many Davis complexes admit nonuniform lattices.

Some lattices are proven to be infinitely generated.

Abstract

Let $\Sigma$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $\Sigma$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin 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 $\Gamma$, and an infinite family of uniform lattices with covolumes converging to that of $\Gamma$. It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice $\Gamma$ is not finitely generated. Examples of $\Sigma$ 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 $\Sigma$.