Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

TL;DR

This paper investigates the structure of normal subgroups within the algebraic fundamental group of affine curves over fields of positive characteristic, revealing their relation to free profinite groups under certain conditions.

Contribution

It establishes that such normal subgroups are isomorphic to subgroups of free profinite groups when their quotients have infinitely generated Sylow p-subgroups.

Findings

Normal subgroups are isomorphic to subgroups of free profinite groups.

Proper open subgroups of these normal subgroups are free profinite groups.

Results apply to affine curves over algebraically closed fields of characteristic p.

Abstract

Let $\pi_1(C)$ be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic $p>0$ of countable cardinality. Let $N$ be a normal (resp. characteristic) subgroup of $\pi_1(C)$. Under the hypothesis that the quotient $\pi_1(C)/N$ admits an infinitely generated Sylow $p$-subgroup, we prove that $N$ is indeed isomorphic to a normal (resp. characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of $N$ is a free profinite group of countable cardinality.