Subgroup structure of fundamental groups in positive characteristic

TL;DR

This paper provides a criterion for when certain closed subgroups of the étale fundamental group of a smooth affine curve over a field of characteristic p are free, using a new version of almost ω-freeness and Haran-Shapiro induction.

Contribution

It introduces a new criterion for profinite freeness of subgroups in positive characteristic and establishes a strong form of almost ω-freeness for the fundamental group.

Findings

Closed subgroups between two normal subgroups are free if they contain most open subgroups of index p.

Established a strong version of almost ω-freeness for the fundamental group.

Applied Haran-Shapiro induction to prove freeness criteria.

Abstract

Let $\Pi$ be the \'etale fundamental group of a smooth affine curve over an algebraically closed field of characteristic $p>0$. We establish a criterion for profinite freeness of closed subgroups of $\Pi$. Roughly speaking, if a closed subgroup of $\Pi$ is "captured" between two normal subgroups, then it is free, provided it contains most of the open subgroups of index $p$. In the proof we establish a strong version of "almost $\omega$-freeness" of $\Pi$ and then apply the Haran-Shapiro induction.