On subgroups of Coxeter groups 

W. Dicks and I. J. Leary

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.  

Geometry and Cohomology in Group Theory London Math. Soc. Lecture Note
Series 252, Cambridge U. P. (1998) 124-160.