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We study the properties of the fundamental group of an affine curve over analgebraically closed field of characteristic $p$, from the point of view ofembedding problems. In characteristic zero, the fundamental group is free, butin characteristic $p$ it is not even $\omega$-free. In this paper we show thatit is "almost $\omega$-free," in the sense that each finite embedding problemhas a proper solution when restricted to some open subgroup. We also prove thatembedding problems can always be properly solved over the given curve ifsuitably many additional branch points are allowed, in locations that can bespecified arbitrarily; this strengthens a result of the first author.

Embedding problems and open subgroups