Linear groups of finite cohomological dimension

Abstract

Our main result provides necessary and sufficient conditions for a finitelygenerated subgroup of GLn(C), n > 0, to have finite virtual cohomological dimension. A group has finite virtual cohomological dimension (VCD) if it has a subgroup of finite index which has finite cohomological dimension; this dimension is, in fact, the same for all torsion-free subgroups of finite index. It is, of course, necessary for a group r with VCD(T) Q), is independent of the choice of solvable series; thus, for a polycyclic group r, h(T) is the number of infinite factors in a normal series with cyclic quotients. We announce our main result.