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Twisted conjugacy and separability

  • Sam Tertooy ORCID logo EMAIL logo
Published/Copyright: December 5, 2024

Abstract

A group 𝐺 is twisted conjugacy separable if, for every automorphism 𝜑, distinct 𝜑-twisted conjugacy classes can be separated in a finite quotient. Likewise, 𝐺 is completely twisted conjugacy separable if, for any group 𝐻 and any two homomorphisms φ , ψ from 𝐻 to 𝐺, distinct ( φ , ψ ) -twisted conjugacy classes can be separated in a finite quotient. We study how these properties behave with respect to taking subgroups, quotients and finite extensions, and compare them to other notions of separability in groups. Finally, we show that, for polycyclic-by-nilpotent-by-finite groups, being completely twisted conjugacy separable is equivalent to all quotients being residually finite.

Acknowledgements

The author is indebted to Peter Wong for bringing the work by Stebe on nests to his attention, to Karel Dekimpe for catching a mistake in the proof of Theorem B and helping to correct said proof, and to the anonymous referee for a multitude of helpful comments and valuable suggestions.

  1. Communicated by: Benjamin Klopsch

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Received: 2024-03-07
Revised: 2024-10-28
Published Online: 2024-12-05
Published in Print: 2025-05-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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