On the birational anabelian program initiated by Bogomolov I

Recall that a program initiated by Bogomolov in 1990 aims at reconstructing function fields K|k with td(K|k)>1 and k algebraically closed from the maximal pro-ℓ abelian-by-central Galois group P^c  _K\Pi ^{\hbox to1pt{}{\rm c}_{\phantom {g}}}_{K} of K, where ℓ is any prime number ≠char(k). In this paper we complete that program in the case k is an algebraic closure of a finite field.