Subgroup Structure of Fundamental Groups in Positive Characteristic

Let Π 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 Π. Roughly speaking, if a closed subgroup of Π 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 ω-freeness" of Π and then apply the Haran-Shapiro induction.