Embedding problems for subgroups of the fundamental group

TL;DR

The paper proves that for the fundamental group of a smooth affine curve over an algebraically closed field of positive characteristic, any embedding problem can be solved on an open subgroup, showing it is nearly -free.

Contribution

It demonstrates that the fundamental group of such a curve is 'almost -free' by solving embedding problems on open subgroups, extending understanding of its structure.

Findings

Existence of open index p normal subgroups solving embedding problems

Fundamental group is 'almost -free'

Extension of structural understanding of fundamental groups in positive characteristic

Abstract

Let $\pi_1(C)$ 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 $\pi_1(C)$ there exist an open index $p$ normal subgroup of $\pi_1(C)$ so that the embedding problem restricted to this group has a solution. In particular, $\pi_1(C)$ is "almost $\omega$-free".