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Twisted conjugacy in residually finite groups of finite Prüfer rank

  • Evgenij Troitsky ORCID logo
Published/Copyright: February 6, 2024

Abstract

Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R ( φ ) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R ( φ ) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ ̂ : [ ρ ] [ ρ φ ] (i.e. we prove the TBFT𝑓, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).

Award Identifier / Grant number: 21-11-00080

Funding statement: The work is supported by the Russian Science Foundation under grant 21-11-00080.

Acknowledgements

The author is indebted to an anonymous reviewer for their careful reading of the paper and very helpful and friendly criticism and suggestions.

  1. Communicated by: Benjamin Klopsch

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Received: 2023-06-03
Revised: 2023-11-21
Published Online: 2024-02-06
Published in Print: 2025-01-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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