Homology and derived p-series of groups

We give homological conditions that ensure that a group homomorphism induces an isomorphism modulo any term of the derived p − series, in analogy to Stallings's 1963 result for the p-lower central series. In fact, we prove a stronger theorem that is analogous to Dwyer's extensions of Stallings’ results. It follows that spaces that are ℤp-homology equivalent have isomorphic fundamental groups modulo any term of their p-derived series. Various authors have related the ranks of the successive quotients of the p-lower central 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 π1(M) are free, to whether finite index subgroups of π1(M) map onto non-abelian free groups, and to whether finite covers of M are ‘large’ in various other senses.