Almost locally free fields
A profinite group \(G\) is called \textit{almost \(\omega\)-free}, if every finite embedding problem \((G\twoheadrightarrow A\twoheadleftarrow B|\) \(A,B\) : finite\()\) has a virtual solution from an open subgroup of \(G\) onto \(B\). A profinite group \(G\) is called \textit{almost locally free}, if for each positive integer \(e\) and for almost all \(e\)-tuple \((\sigma_1,\dots,\sigma_e)\in G^e\) in the sense of the Haar measure, the closed subgroup generated by \(\sigma_1,\dots,\sigma_e\) is free on \(e\)-generators. It is shown that an almost locally free profinite group is almost \(\omega\)-free. Using the generalized Abhyankar's conjecture, the author deduces that the fundamental group of the smooth connected affine curve \(C\) over an algebraically closed field of positive characteristic is almost locally free, hence is almost \(\omega\)-free.
scientific article; zbMATH DE number 5994284