ON GROUPS OF TYPE (FP) INFINITY

Let G be a group. A ZG-module M is said to be of type (FP)infinity over ZG if and only if there is a projective resolution P* -->> M in which every P(i) is finitely generated. We show that if G belongs to a large class of torsion-free groups, which includes torsion-free linear and soluble-by-finite groups, then every ZG-module of type (FP)infinity has finite projective dimension. We also prove that every soluble or linear group of type (FP)infinity is virtually of type (FP). The arguments apply to groups which admit hierarchical decompositions. We also make crucial use of a generalized theory of Tate cohomology recently developed by Mislin.