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.