Abstract
We have already pointed out that a profinite group G of an infinite rank m is free of rank m if (and only if) G is projective and every finite split embedding problem for G with a nontrivial kernel has m solutions (Proposition 9.4.7). Dropping the condition on G to be projective leads to the notion of a “quasi-free profinite group” (Section 10.6).
A somewhat stronger condition is that of a “semi-free profinite group”. We say that G is semi-free if every finite split embedding problem for G with a nontrivial kernel has m independent solutions (Definition 10.1.5). The advantage of the latter notion on the former one is that the known conditions on a closed subgroup of a free profinite group of rank m to be free of rank m go over to semi-free groups. Indeed, even the method of proof that applies twisted wreath products goes over from free profinite groups to semi-free profinite groups (Section 10.3). Thus, every open subgroup of a semi-free group G is semi-free (Lemma 10.4.1), every normal closed subgroup N of G with G/N Abelian is semi-free, every proper open subgroup of a closed normal subgroup of G is semi-free, and in general every closed subgroup M of G that is “contained in a diamond” is semi-free (Theorem 10.5.3).
An application of the diamond theorem to function fields of one variable over PAC fields appears in the next chapter.
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© 2011 Springer-Verlag Berlin Heidelberg
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Jarden, M. (2011). Semi-Free Profinite Groups. In: Algebraic Patching. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15128-6_10
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DOI: https://doi.org/10.1007/978-3-642-15128-6_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15127-9
Online ISBN: 978-3-642-15128-6
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