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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Virtual retraction and Howson’s theorem in pro-$p$ groups
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by Mark Shusterman and Pavel Zalesskii PDF
Trans. Amer. Math. Soc. 373 (2020), 1501-1527 Request permission

Abstract:

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 $\tau \colon U \to K$ 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.
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Additional Information
  • Mark Shusterman
  • Affiliation: Raymond and Beverly Sackler School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel
  • MR Author ID: 1157328
  • Email: markshus@mail.tau.ac.il
  • Pavel Zalesskii
  • Affiliation: Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília DF, Brazil
  • MR Author ID: 245312
  • Email: pz@mat.unb.br
  • Received by editor(s): October 6, 2017
  • Received by editor(s) in revised form: July 18, 2018
  • Published electronically: December 4, 2019
  • Additional Notes: The first author was partially supported by a grant of the Israel Science Foundation with cooperation of UGC no. 40/14. The first author is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship.
    The second author was partially supported by CAPES and CNPq.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 1501-1527
  • MSC (2010): Primary 11520, 20B07, 20E18, 20F65, 20F69
  • DOI: https://doi.org/10.1090/tran/7784
  • MathSciNet review: 4068271