Homological stability of series of groups

TL;DR

This paper introduces a new framework for understanding the stability of group series under homology equivalences, unifying various classical theorems and extending them to broader contexts.

Contribution

It proposes the concept of subgroup stability under classes of maps, providing a unified approach to existing theorems and comparing it with homological localization.

Findings

Unified framework for stability of group series

Extension of classical theorems to new contexts

Comparison with homological localization

Abstract

``What aspects of a group are unchanged, or stable, under homology equivalences''? The model theorem in this regard is the 1963 result of J. Stallings that the lower central series is preserved under any integral homological equivalence of groups. Various other theorems of this nature have since appeared. Stallings himself proved similar theorems for homology with rational or mod p coefficients. These involved different series of groups- variations of the lower central series. W. Dwyer generalized Stallings' integral results to larger classes of maps, work that was completed in the other cases by the authors. More recently the authors proved analogues of the theorems of Stallings and Dwyer for variations of the derived series. The above theorems are all different but clearly have much in common. Here we present a new concept, that of the stability of a subgroup, or a series of subgroups under a class of maps, that offers a framework in which all of these theorems can be viewed. We contrast it with homological localization of groups, which is a previously well-studied framework that might also be applied to these questions.