Embedding problems and open subgroups

Harbater, David; Stevenson, Katherine

doi:10.1090/S0002-9939-2010-10776-1

Embedding problems and open subgroups
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by David Harbater and Katherine Stevenson PDF

Proc. Amer. Math. Soc. 139 (2011), 1141-1154 Request permission

Abstract:

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.

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