Almost locally free fields

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Using the positive solution of the general Abhyankar's 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)επ1(C)e 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.

Original languageEnglish
Pages (from-to)171-176
Number of pages6
JournalJournal of Algebra
Issue number1
StatePublished - 1 Jun 2011


  • Abhyankar's conjecture
  • Embedding problems
  • Galois group
  • ω-free groups


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