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Twisted conjugacy and separability

  • Sam Tertooy ORCID logo EMAIL logo
Veröffentlicht/Copyright: 5. Dezember 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

References

[1] R. B. J. T. Allenby and R. J. Gregorac, On locally extended residually finite groups, Conference on Group Theory, Lecture Notes in Math. 319, Springer, Berlin (1973), 9–17. 10.1007/BFb0058924Suche in Google Scholar

[2] J. Deré and M. Pengitore, Effective twisted conjugacy separability of nilpotent groups, Math. Z. 292 (2019), no. 3–4, 763–790. 10.1007/s00209-018-2102-5Suche in Google Scholar

[3] A. Fel’shtyn and B. Klopsch, Pólya-Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions, Indag. Math. (N. S.) 33 (2022), no. 4, 753–767. 10.1016/j.indag.2022.02.004Suche in Google Scholar

[4] A. Fel’shtyn and E. Troitsky, Twisted conjugacy separable groups, preprint (2006), https://arxiv.org/abs/math/0606764. Suche in Google Scholar

[5] A. Fel’shtyn and E. Troitsky, Twisted Burnside–Frobenius theory for discrete groups, J. Reine Angew. Math. 613 (2007), 193–210. 10.1515/CRELLE.2007.097Suche in Google Scholar

[6] E. Formanek, Conjugate separability in polycyclic groups, J. Algebra 42 (1976), no. 1, 1–10. 10.1016/0021-8693(76)90021-1Suche in Google Scholar

[7] D. Gonçalves and P. Wong, Twisted conjugacy classes in exponential growth groups, Bull. Lond. Math. Soc. 35 (2003), no. 2, 261–268. 10.1112/S0024609302001832Suche in Google Scholar

[8] D. L. Gonçalves and P. N.-S. Wong, Homogeneous spaces in coincidence theory. II, Forum Math. 17 (2005), no. 2, 297–313. 10.1515/form.2005.17.2.297Suche in Google Scholar

[9] A. V. Goryaga, Example of a finite extension of an FAC-group that is not an FAC-group, Sibirsk. Mat. Zh. 27 (1986), no. 3, 203–205. Suche in Google Scholar

[10] A. V. Jategaonkar, Integral group rings of polycyclic-by-finite groups, J. Pure Appl. Algebra 4 (1974), 337–343. 10.1016/0022-4049(74)90013-9Suche in Google Scholar

[11] S. C. Jeanes and J. S. Wilson, On finitely generated groups with many profinite-closed subgroups, Arch. Math. (Basel) 31 (1978/79), no. 2, 120–122. 10.1007/BF01226424Suche in Google Scholar

[12] D. Kilsch, Conjugacy separability and separable orbits, J. Pure Appl. Algebra 30 (1983), no. 2, 167–179. 10.1016/0022-4049(83)90053-1Suche in Google Scholar

[13] L. A. Kurdachenko and J. Otal, FC-groups all of whose factor groups are residually finite, Comm. Algebra 31 (2003), no. 3, 1235–1251. 10.1081/AGB-120017764Suche in Google Scholar

[14] A. I. Mal’cev, On homomorphisms onto finite groups, Ivanov. Gos. Ped. Inst. Učen. Zap. 18 (1958), 49–60. Suche in Google Scholar

[15] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb. (3) 17, Springer, Berlin, 1991. 10.1007/978-3-642-51445-6Suche in Google Scholar

[16] A. Martino and A. Minasyan, Conjugacy in normal subgroups of hyperbolic groups, Forum Math. 24 (2012), no. 5, 889–910. 10.1515/form.2011.089Suche in Google Scholar

[17] M. Menth, Nilpotent groups with every quotient residually finite, J. Group Theory 5 (2002), no. 2, 199–217. 10.1515/jgth.5.2.199Suche in Google Scholar

[18] N. Nikolov and D. Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds, Ann. of Math. (2) 165 (2007), no. 1, 171–238. 10.4007/annals.2007.165.171Suche in Google Scholar

[19] N. Nikolov and D. Segal, Generators and commutators in finite groups; abstract quotients of compact groups, Invent. Math. 190 (2012), no. 3, 513–602. 10.1007/s00222-012-0383-6Suche in Google Scholar

[20] V. N. Remeslennikov, Conjugacy in polycyclic groups, Algebra Logika 8 (1969), 712–725. 10.1007/BF02219654Suche in Google Scholar

[21] L. Ribes and P. Zalesskii, Profinite Groups, 2nd ed., Ergeb. Math. Grenzgeb. (3) 40, Springer, Berlin, 2010. 10.1007/978-3-642-01642-4Suche in Google Scholar

[22] D. J. S. Robinson, A. Russo and G. Vincenzi, On groups whose subgroups are closed in the profinite topology, J. Pure Appl. Algebra 213 (2009), no. 4, 421–429. 10.1016/j.jpaa.2008.07.015Suche in Google Scholar

[23] D. J. S. Robinson, A. Russo and G. Vincenzi, On the theory of generalized FC-groups, J. Algebra 326 (2011), 218–226. 10.1016/j.jalgebra.2009.04.002Suche in Google Scholar

[24] D. M. Smirnov, On the theory of finitely approximable groups, Ukrain. Mat. Ž. 15 (1963), 453–457. Suche in Google Scholar

[25] P. F. Stebe, Conjugacy separability of groups of integer matrices, Proc. Amer. Math. Soc. 32 (1972), 1–7. 10.2307/2038292Suche in Google Scholar

[26] P. F. Stebe, Nests in nilpotent groups, Houston J. Math. 2 (1976), no. 3, 419–426. Suche in Google Scholar

[27] P. F. Stebe, A residual property of certain linear groups, Trans. Amer. Math. Soc. 302 (1987), no. 1, 333–340. 10.2307/2000913Suche in Google Scholar

[28] E. V. Troitskiĭ, Two examples related to twisted Burnside–Frobenius theory for infinitely generated groups, J. Math. Sci. 248 (2020), no. 5, 661–666. 10.1007/s10958-020-04903-0Suche in Google Scholar

[29] B. A. F. Wehrfritz, Two examples of soluble groups that are not conjugacy separable, J. Lond. Math. Soc. (2) 7 (1973), 312–316. 10.1112/jlms/s2-7.2.312Suche in Google Scholar

[30] B. A. F. Wehrfritz, Another example of a soluble group that is not conjugacy separable, J. Lond. Math. Soc. (2) 14 (1976), no. 2, 381–382. 10.1112/jlms/s2-14.2.381Suche in Google Scholar

[31] B. A. F. Wehrfritz, Linear groups with all subgroups profinitely closed, Q. J. Math. 62 (2011), no. 2, 501–512. 10.1093/qmath/haq004Suche in Google Scholar

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

Heruntergeladen am 27.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jgth-2024-0056/pdf?lang=de
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