The Bogomolov-Pop conjecture: function fields over algebraically closed fields are determined by their pro-l abelian-by-central Galois groups

Theorem 1.1 (principal paper): for function fields $K/k$, $L/l$ of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from $\ell$, the map from isomorphisms of perfect closures modulo Frobenius to bracket-compatible isomorphisms $\Pi_L^a\to\Pi_K^a$ modulo $\mathbb Z_\ell^\times$ is bijective; equal characteristic and transcendence degree are forced. Companions: the same reconstruction from $K_1^M/\ell$, $K_2^M/\ell$ and their product (removing Topaz's rational-subgroup data and transcendence degree at least five), and from $K_1^M/p$ with Steinberg relations in characteristic $p$. Not shown: transcendence degree one, or fields with non-algebraically-closed constants.

Theorem 1.1 (principal paper): for function fields $K/k$, $L/l$ of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from $\ell$, the map from isomorphisms of perfect closures modulo Frobenius to bracket-compatible isomorphisms $\Pi_L^a\to\Pi_K^a$ modulo $\mathbb Z_\ell^\times$ is bijective; equal characteristic and transcendence degree are forced. Companions: the same reconstruction from $K_1^M/\ell$, $K_2^M/\ell$ and their product (removing Topaz's rational-subgroup data and transcendence degree at least five), and from $K_1^M/p$ with Steinberg relations in characteristic $p$. Not shown: transcendence degree one, or fields with non-algebraically-closed constants.