Commuting-Liftable Subgroups of Galois Groups

Let n denote either a positive integer or ∞, let ell be a fixed prime and let K be a field of characteristic different from ell. In the presence of sufficiently many roots of unity, we show how to recover much of the decomposition/inertia structure of valuations in the Z/elln -elementary abelian Galois group of K, while using only the group-theoretical structure of the Z/ellN-abelian-by-central Galois group of K whenever N is sufficiently large with respect to n. Moreover, if n = 1 then N = 1 suffices, while if n neq ∞, we provide an explicit N0 neq ∞, as a function of n and ell, for which all N ≥ N0 suffice above. In the process, we give a complete classification of so-called "commuting-liftable subgroups" of elementary-abelian Galois groups and prove that they always arise from valuations.