We study the properties of the fundamental group of an affine curve over an algebraically closed field of characteristic $p$, from the point of view of embedding problems. In characteristic zero, the fundamental group is free, but in characteristic $
p$ it is not even $omega$-free. In this paper we show that it is almost $omega$-free, in the sense that each finite embedding problem has a proper solution when restricted to some open subgroup. We also prove that embedding problems can always be properly solved over the given curve if suitably many additional branch points are allowed, in locations that can be specified arbitrarily; this strengthens a result of the first author.
We prove a generalization of Shafarevichs Conjecture for fields of Laurent series in two variables over an arbitrary field. While not projective, the absolute Galois group of such a field is shown to be semi-free. We also show that the function field
of a smooth projective curve over a large field has semi-free absolute Galois group. In the first edition of this paper it was shown that these groups are quasi-free, which is weaker.
