Homological Stability of Series of Groups

We define the stability of a subgroup under a class of maps, and establish the basic properties of this notion. Loosely speaking, we will say that a normal subgroup, or more generally a normal series {An} of a group A, is stable under a class of homomorphisms H if whenever f : A → B lies in H, we have that f(a) ∈ Bn if and only if a ∈ An. This translates to saying that each element of H induces a monomorphism A/An →֒ B/Bn. This contrasts with the usual theories of localization wherein one is concerned with situations where f induces an isomorphism. In the literature, the most commonly considered class of maps are those that induce isomorphisms on (low-dimensional) group homology. The model theorem in this regard is the 1963 result of J. Stallings that (each term of) the lower central series is preserved under any Z-homological equivalence of groups [9]. Various other theorems of this nature have since appeared, involving different series of groupsvariations of the lower central series. W. Dwyer generalized Stallings’ Z results to larger classes of maps [7], 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 [4] [6] [5]. The above theorems are all different but clearly have much in common. We interpret all of these results in the framework of stability.