Homology and Derived p-Series of Groups

Abstract

We prove that groups that are mod-p-homology equivalent are isomorphic modulo any term of their derived p-series, in precise analogy to Stallings' 1963 result for the lower-central p-series. Similarly spaces that are mod-p-homology equivalent have fundamental groups that are isomorphic modulo any term of their p-derived series. Various authors have related the ranks of the successive quotients of the lower central p-series and of the derived p-series of the fundamental group of a 3-manifold M to the volume of M, to whether certain subgroups of the fundamental group of M are free, whether finite index subgroups of the fundamental group of M map onto non-abelian free groups, and to whether finite covers of M are ``large'' in various other senses. Specifically, let A be a finitely-generated group and B be a finitely presented group. If a homomorphism induces an isomorphism (respectively monomorphism) on H_1(- ;Z_p) and an epimorphism on H_2(- ;Z_p), then for each finite n, it induces an isomorphism (respectively monomorphism) between the quotients of A and B by the n-th terms of their respective p-derived series. In fact we prove a stronger version that is analogous to Dwyer's extension of Stallings' theorem.