Homology and Derived P-series of Groups

We prove that groups that are Zp-homology equivalent are isomorphic modulo any term of their derived p-series, in precise analogy to Stallings’ 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. Similarly, 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, 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.