Simple Lattices in Products of Davis Complexes

TL;DR

This paper constructs simple uniform lattices in products of Davis complexes, extending Burger and Mozes' work on trees to more complex geometric structures.

Contribution

It introduces a new class of simple lattices in Davis complexes and defines an analogue of Burger-Mozes universal groups for these settings.

Findings

Constructed simple uniform lattices in Davis complexes.

Defined an analogue of Burger-Mozes universal groups.

Provided a local density criterion for vertex transitive groups.

Abstract

Burger and Mozes (1997) constructed the first examples of simple uniform lattices in products of trees. In this paper, we construct simple uniform lattices in products of certain Davis complexes. More precisely, we consider lattices in products of trees and two-dimensional Davis complexes of the right-angled Coxeter group whose defining graph is an odd graph. As part of the proof, we define an analogue of the Burger-Mozes universal groups in this setting, and provide a local criterion for a vertex transitive group to be dense in the universal group.