  The aim of Bogomolov's programme is to prove birational anabelian conjectures
for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically
closed fields. The present article is concerned with the 1-dimensional case.
While it is impossible to recover $K|k$ from its absolute Galois group alone,
we prove that it can be recovered from the pair
$(\mathrm{Aut}(\overline{K}|k),\mathrm{Aut}(\overline{K}|K))$, consisting of
the absolute Galois group of $K$ and the larger group of field automorphisms
fixing only the base field.
