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

Abstract

Let π1(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 (respectively, characteristic) subgroup of π 1(C). Under the hypothesis that the quotient π 1(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (respectively, 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.