On subgroups of Coxeter groups

TL;DR

This paper explores the homological properties of Coxeter groups, constructing examples with unusual cohomological dimensions and providing minimal presentations for certain subgroups, advancing understanding of their algebraic and topological structure.

Contribution

It introduces new constructions of torsion-free groups with specific cohomological properties and characterizes Coxeter groups with finite virtual cohomological dimension.

Findings

Constructed torsion-free groups with different cohomological dimensions over various rings.

Determined which Coxeter groups have finite virtual cohomological dimension.

Provided minimal presentations for certain torsion-free finite-index subgroups.

Abstract

A right-angled Coxeter group is a group with a given set of generators of order two, subject only to the relations that certain pairs of the generators commute. Various papers have shown how homological properties of the Coxeter group are related to homological properties of the simplicial complex whose simplices are the sets of commuting generators. Using these techniques, we construct torsion-free groups which are Poincare duality groups over some rings but not over others, and a group whose integral cohomological dimension is finite but strictly greater than its cohomological dimension over any field. We determine which Coxeter groups have finite virtual cohomological dimension (it is classical that all finitely generated Coxeter groups have finite vcd, but there are others). We also give minimal presentations for certain torsion-free finite-index subgroups of right-angled Coxter groups. Finally we give a `bare-hands' construction (using free products with amalgamation and HNN extensions) of a torsion-free group whose integral cohomological dimension is strictly greater than its rational cohomological dimension.