Homological Finiteness Conditions for Modules Over Group Algebras

We develop a theory of modules of type FP∞ over group algebras of hierarchically decomposable groups. This class of groups is denoted hF and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an hF-group and M a ZG-module of type FP∞ then M has finite projective dimension over ZH for all torsion-free subgroups H of G. We also show that if G is an hF-group of type FP∞ and M is a ZG-module which is ZF-projective for all finite subgroups F of G, then M has finite projective dimension over ZG. Both of these results have as a special case the striking fact that if G is an hF-group of type FP∞ then the torsion-free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite hF-group of type FP∞ has finite virtual cohomological dimension.