Maximal torsion-free subgroups of certain lattices of hyperbolic buildings and Davis complexes

TL;DR

This paper constructs explicit maximal torsion-free subgroups of Coxeter groups acting on hyperbolic buildings and Davis complexes, revealing their structure as amalgams of surface groups over free groups.

Contribution

It provides a new explicit method to find maximal torsion-free subgroups of certain Coxeter groups via polygonal complexes and embeddings into lattices of hyperbolic buildings.

Findings

Constructed explicit maximal torsion-free subgroups.

Embedded fundamental groups into cocompact lattices of hyperbolic buildings.

Identified the structure as amalgams of surface groups over free groups.

Abstract

We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.