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Virtual retraction and Howson’s theorem in pro-\(p\) groups. (English) Zbl 1480.20077

Summary: We show that for every finitely generated closed subgroup \(K\) of a nonsolvable Demushkin group \(G\), there exists an open subgroup \(U\) of \(G\) containing \(K\) and a continuous homomorphism \(\tau : U \to K\) satisfying \(\tau (k) = k\) for every \(k \in K\). We prove that the intersection of a pair of finitely generated closed subgroups of a Demushkin group is finitely generated (giving an explicit bound on the number of generators). Furthermore, we show that these properties of Demushkin groups are preserved under free pro-\( p\) products and deduce that Howson’s theorem holds for the Sylow subgroups of the absolute Galois group of a number field. Finally, we confirm two conjectures of Ribes, thus classifying the finitely generated pro-\( p\) M. Hall groups.

MSC:

20E18 Limits, profinite groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65 Geometric group theory
20F69 Asymptotic properties of groups
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