Almost locally free fields

Using the positive solution of the general Abhyankarʼs conjecture, we prove that the fundamental group π1(C) of the smooth connected affine curve C is “almost free”. That is, for each positive integer e and for almost all σ=(σ1,…,σe)∈π1e(C) in the sense of the Haar measure, the closed subgroup of π1(C) generated by σ1,…,σe is profinite free on e generators. This implies a theorem of Harbater–Stevenson, proved by other means, that every finite embedding problem for π1(C) is solvable, if we restrict the problem to a suitable open subgroups.