  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.
