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Transactions of the American Mathematical Society

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

 
 

 

Virtual retraction and Howson’s theorem in pro-$p$ groups


Authors: Mark Shusterman and Pavel Zalesskii
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
Published electronically: December 4, 2019
MathSciNet review: 4068271
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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.
Article copyright: © Copyright 2019 American Mathematical Society