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Embedding problems for subgroups of the fundamental group

Let $π_1(C)$ be the fundamental group of a smooth irreducible affine curve $C$ over an algebraically closed field of positive characteristic. It is shown that given an embedding problem for $π_1(C)$ there exist an open index $p$ normal subgroup of $π_1(C)$ so that the embedding problem restricted to this group has a solution. In particular, $π_1(C)$ is "almost $ω$-free".