Let \(l\) be a fixed prime number, and \(K\) a field of characteristic \(\operatorname{char}(K) \neq l\). The first key step in most strategies towards anabelian geometry is to develop a local theory allowing to recover inertia and/or decomposition groups of “points” using the given Galois-theoretic information. In the context of algebraic curves, one should eventually detect inertia/decomposition groups of the given curve within its étale fundamental group. On the other hand, in the birational setting, this corresponds to detecting inertia/ decomposition groups of arithmetically and/or geometrically meaningful places of the considered function field within its absolute Galois group. The earliest example of such a local theory is Neukirch’s group-theoretic characterization of decomposition groups of finite places of global fields.{ } Two non-arithmetically based methods were proposed at about the same time for recovering inertial and decomposition groups of valuations using Galois groups. The first one relies on the theory of rigid elements introduced by R. Ware [Can. J. Math. 33, 1338–1355 (1981; Zbl 0514.10015)] and developed by Arason-Elman-Jacob [J. Kr. Arason et al., J. Algebra 110, 449–467 (1987; Zbl 0629.10016)], A. J. Engler and J. B. Nogueira [J. Algebra 166, No. 3, 481–505 (1994; Zbl 0809.12004)], J. Koenigsmann (e.g., [J. Reine Angew. Math. 465, 165–182 (1995; Zbl 0824.12006)]) and I. Efrat (e.g., [Pac. J. Math. 226, No. 2, 259–275 (2006; Zbl 1161.19002)]), among others. The second method is the theory of commuting-liftable pairs (CL-pairs) in Galois groups introduced by Bogomolov and developed by himself and Tschinkel. Its input is the much smaller pro-\(l\) abelian-by-central Galois group, but it requires that the base field contain an algebraically closed subfield. Motivation for the development of this theory primarily comes from the Bogomolov-Pop conjecture providing a precise formulation of Bogomolov’s program for “reconstructing” higher-dimensional function fields over an algebraically closed field from their pro-\(l\) abelian-by-central Galois groups (see [F. A. Bogomolov, Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 1, 32–67 (1991; Zbl 0736.12004)], and [F. Pop, Invent. Math. 187, No. 3, 511–533 (2012; Zbl 1239.14025)]).{ }Until now, the two non-arithmetically based methods remained almost fully separate (although a connection between them has been suggested in the \(l^n\)-abelian-by-central situation by several authors). The paper under review provides an approach that unifies the two methods. Let \(n > 0\) be an integer or equal to infinity. Then the author shows that, for each sufficiently large integer \(N\) and any field \(K\) containing a primitive root of unity of degree \(l^{2l^{N}}\), there is a group-theoretical recipe which recovers (minimized) inertia and decomposition subgroups in the maximal \(l^n\)-elementary abelian Galois group of \(K\). This is obtained by using the structure of the \(l^{N}\)-abelian-by-central Galois group of \(K\). Moreover, if \(n\) is finite, then \(N\) is determined explicitly as well; when \(n = 1\), one may take \(N = 1\).{ }The paper under review obtains analogous results to those in the theory of commuting-liftable pairs, for the \(l^n\)-abelian-by-central and the pro-\(l\) abelian-by-central situations. For this purpose, the author applies the theory of rigid elements, while working under far less restrictive assumptions than needed by the Bogomolov and Tschinkel approach. Thus, Theorems 1 and 2 of the present paper yield a non-trivial generalization of results of [F. Bogomolov and Y. Tschinkel, Int. Press Lect. Ser. 3, No. I, 75–120 (2002; Zbl 1048.11090)]. As an application, the paper exhibits several important classes of function fields of zero characteristic whose absolute Galois groups are not realizable as absolute Galois groups of fields of nonzero characteristic.