  Let $\Pi^c_K\to\Pi_K$ be the maximal pro-$\ell$ abelian-by-central,
respectively abelian, Galois groups of a function field $K|k$ with $k$
algebraically closed and ${\rm char}\neq\ell$. We show that $K|k$ can be
functorially reconstructed by group theoretical recipes from $\Pi^c_K$ endowed
with the set of divisorial inertia ${\rm Inrdiv}(K)\subset\Pi_K$. As
applications, one has: (i) A group theoretical recipe to reconstruct $K|k$ from
$\Pi^c_K$, provided either ${\rm Tr.deg}(K|k)>{\rm dim}(k)+1$ or ${\rm
tr.deg}(K|k) >{\rm dim}(k)>1$, where ${\rm dim}(k)$ is the Kronecker dimension;
(ii) An application to the pro-$\ell\!$ abelian-by-central I/OM (Ihara's
question / Oda-Matsumoto conjecture), which in the cases considered here
implies the classical I/OM.
