Virtual retraction and Howson's theorem in pro-p groups

2017/05/25 by Shusterman, Mark, Zalesskii, Pavel
#11S20 #20B07 #20E18 #20F65 #20F69 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1705.09096

Abstract

We show that for every finitely generated closed subgroup K of a non-solvable Demushkin group G, there exists an open subgroup U of G containing K, and a continuous homomorphism τ\colon U → K satisfying τ(k) = k for every k ∈ 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.

Related

• Demushkin groups and inverse Galois theory for pro-p-groups of finite rank and maximal p-extensions
• The pro-k-solvable topology on a free group
• Virtual retraction properties in groups
• Free product of Demushkin groups as absolute Galois group
• Pro-p groups with constant generating number on open subgroups