Subgroups of profinite surface groups

TL;DR

This paper investigates the subgroup structure of the étale fundamental group of a projective curve over an algebraically closed field of characteristic zero, revealing that most infinite index normal subgroups are semi-free and establishing an analog of the diamond theorem.

Contribution

It introduces an analog of the diamond theorem for the étale fundamental group and demonstrates that most infinite index normal subgroups are semi-free, extending subgroup theory in algebraic geometry.

Findings

Most normal subgroups of infinite index are semi-free

Proper open subgroups of such normal subgroups are semi-free

Establishes an analog of the diamond theorem for the fundamental group

Abstract

We study the subgroup structure of the \'etale fundamental group $\Pi$ of a projective curve over an algebraically closed field of characteristic 0. We obtain an analog of the diamond theorem for $\Pi$. As a consequence we show that most normal subgroups of infinite index are semi-free. In particular every proper open subgroup of a normal subgroup of infinite index is semi-free.