Embedding problems and open subgroups

TL;DR

This paper investigates the fundamental group of affine curves over algebraically closed fields of characteristic p, showing it is nearly ω-free and can solve embedding problems with added branch points.

Contribution

It proves the fundamental group is almost ω-free and demonstrates solutions to embedding problems with additional branch points, extending previous results.

Findings

Fundamental group is almost ω-free in characteristic p.

Embedding problems can be solved with added branch points.

Strengthens previous results on embedding problems.

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.