Free absolute Galois groups
David Harbater

By joint work with K. Stevenson, a profinite group is free iff it is projective and quasi-free (in the sense that every finite split embedding problem has "enough" solutions). Using this, the absolute Galois group of K is shown to be free in the following two cases: (i) K is the function field of a real curve with no real points; (ii) K is the maximal abelian extension of k((x,y)), where k is algebraically closed of characteristic 0. Here the absolute Galois group is shown to be quasi-free using geometric methods, and the projectivity follows from work of Colliot-Thelene, Parimala and Ojanguren.