Subgroup structure of fundamental groups in positive characteristic

Abstract

Let $\Pi$ be the étale 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.