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 ℤ p ‐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 π 1 ( M ) are free, to whether finite index subgroups of π 1 ( M ) map onto non‐abelian free groups, and to whether finite covers of M are ‘large’ in various other senses.