Abstract: Let $K$ be a field and $\ell$ be a prime such that $\operatorname{char} K \neq \ell$. In the presence of sufficiently many roots of unity in $K$, we show how to recover some of the inertia/decomposition structure of valuations inside the maximal $(\mathbb{Z}/\ell)$-abelian resp. pro-$\ell$-abelian Galois group of $K$ using its $(\mathbb{Z}/\ell)$-central resp. pro-$\ell$-central extensions.

Note: The results in this paper were subsumed by combining the the results of the following more comprehensive papers: Commuting-Liftable Subgroups of Galois Groups II and Abelian-by-central Galois groups I: a formal description