Skip to main content
SpringerLink
Log in

Two Examples Related to the Twisted Burnside–Frobenius Theory for Infinitely Generated Groups

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

TheTBFTf conjecture, which is a modification of a conjecture by Fel’shtyn and Hill, says that if the Reidemeister number R(𝜙) of an automorphism 𝜙 of a (countable discrete) group G is finite, then it coincides with the number of fixed points of the corresponding homeomorphism \( \hat{\phi} \) of \( {\hat{G}}_f \) (the part of the unitary dual formed by finite-dimensional representations). The study of this problem for residually finite groups has been the subject of some recent activity. We prove here that for infinitely generated residually finite groups there are positive and negative examples for this conjecture. It is detected that the finiteness properties of the number of fixed points of 𝜙 itself also differ from the finitely generated case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Bleak, A. Fel’shtyn, and D. L. Gonçalves, “Twisted conjugacy classes in R. Thompson’s group F,” Pacific J. Math., 238, No. 1, 1–6 (2008).

  2. J. Burillo, F. Matucci, and E. Ventura, The Conjugacy Problem in Extensions of Thompson’s Group F, arXiv:1307.6750 (2013).

  3. K. Dekimpe and D. Gonçalves, “The R∞ property for free groups, free nilpotent groups and free solvable groups,” Bull. London Math. Soc., 46, No. 4, 737–746 (2014).

  4. K. Dekimpe and D. Gonçalves, “The R∞ property for Abelian groups,” Topol. Methods Nonlinear Anal., 46, No. 2, 773–784 (2015).

  5. K. Dekimpe and P. Penninckx, “The finiteness of the Reidemeister number of morphisms between almost-crystallographic groups,” J. Fixed Point Theory Appl., 9, No. 2, 257–283 (2011).

    Article  MathSciNet  Google Scholar 

  6. A. Fel’shtyn, Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion, Mem. Am. Math. Soc., Vol. 147, No. 699 (2000).

  7. A. Fel’shtyn, “The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite,” Zap. Nauch. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 279, No. 6, 229–240, 250 (2001).

  8. A. Fel’shtyn and R. Hill, “The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion,” K-Theory, 8, No. 4, 367–393 (1994).

  9. A. Fel’shtyn, Yu. Leonov, and E. Troitsky, “Twisted conjugacy classes in saturated weakly branch groups,” Geom. Dedicata, 134, 61–73 (2008).

    Article  MathSciNet  Google Scholar 

  10. A. Fel’shtyn and E. Troitsky, “A twisted Burnside theorem for countable groups and Reidemeister numbers,” in: C. Consani and M. Marcolli, eds., Noncommutative Geometry and Number Theory, Vieweg, Braunschweig (2006), pp. 141–154.

    Chapter  Google Scholar 

  11. A. Fel’shtyn and E. Troitsky, “Twisted Burnside–Frobenius theory for discrete groups,” J. Reine Angew. Math., 613, 193–210 (2007).

    MathSciNet  MATH  Google Scholar 

  12. A. Fel’shtyn, E. Troitsky, and A. Vershik, “Twisted Burnside theorem for type II1 groups: an example,” Math. Res. Lett., 13, No. 5, 719–728 (2006).

    Article  MathSciNet  Google Scholar 

  13. A. Fel’shtyn, “New directions in Nielsen–Reidemeister theory,” Topology Appl., 157, No. 10-11, 1724–1735 (2010).

    Article  MathSciNet  Google Scholar 

  14. A. Fel’shtyn and D. L. Gonçalves, “The Reidemeister number of any automorphism of a Baumslag–Solitar group is infinite,” in: Geometry and Dynamics of Groups and Spaces, Progr. Math., Vol. 265, Birkhäuser, Basel (2008), pp. 399–414.

  15. A. Fel’shtyn and D. L. Gonçalves, “Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups,” Geom. Dedicata, 146, 211–223 (2010), with an appendix written jointly with Francois Dahmani.

  16. A. Fel’shtyn and D. L. Gonçalves, “Reidemeister spectrum for metabelian groups of the form Qn ⋊ ℤ and ℤ[1/p]n ⋊ ℤ, p prime,” Internat. J. Algebra Comput., 21, No. 3, 505–520 (2011).

  17. A. Fel’shtyn, N. Luchnikov, and E. Troitsky, “Twisted inner representations,” Russ. J. Math. Phys., 22, No. 3, 301–306 (2015).

    Article  MathSciNet  Google Scholar 

  18. A. Fel’shtyn and T. Nasybullov, “The R∞ and S∞ properties for linear algebraic groups,” J. Group Theory, 19, No. 5, 901–921 (2016).

    Article  MathSciNet  Google Scholar 

  19. A. Fel’shtyn and E. Troitsky, Twisted Conjugacy Classes in Residually Finite Groups, arXiv:1204. 3175 (2012).

  20. A. Fel’shtyn and E. Troitsky, “Aspects of the property R∞,” J. Group Theory, 18, No. 6, 1021–1034 (2015).

    MathSciNet  MATH  Google Scholar 

  21. D. Gonçalves and D. H. Kochloukova, “Sigma theory and twisted conjugacy classes,” Pacific J. Math., 247, No. 2, 335–352 (2010).

    Article  MathSciNet  Google Scholar 

  22. D. Gonçalves and P. Wong, “Twisted conjugacy classes in wreath products,” Internat. J. Algebra Comput., 16, No. 5, 875–886 (2006).

    Article  MathSciNet  Google Scholar 

  23. D. Gonçalves and P. Wong, “Twisted conjugacy classes in nilpotent groups,” J. Reine Angew. Math., 633, 11–27 (2009).

    MathSciNet  MATH  Google Scholar 

  24. L. Guyot and Y. Stalder, “Limits of Baumslag–Solitar groups and dimension estimates in the space of marked groups,” Groups Geom. Dyn., 6, No. 3, 533–577 (2012).

    Article  MathSciNet  Google Scholar 

  25. E. Jabara, “Automorphisms with finite Reidemeister number in residually finite groups,” J. Algebra, 320, No. 10, 3671–3679 (2008).

    Article  MathSciNet  Google Scholar 

  26. B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math., Vol. 14, Amer. Math. Soc., Providence (1983).

  27. A. Juhász, “Twisted conjugacy in certain Artin groups,” in: Ischia Group Theory 2010, eProceedings, World Scientific (2011), pp. 175–195.

  28. G. Levitt and M. Lustig, “Most automorphisms of a hyperbolic group have very simple dynamics,” Ann. Scient. Éc. Norm. Sup., 33, 507–517 (2000).

    Article  MathSciNet  Google Scholar 

  29. T. Mubeena and P. Sankaran, “Twisted conjugacy classes in Abelian extensions of certain linear groups,” Can. Math. Bull., 57, No. 1, 132–140 (2014).

    Article  MathSciNet  Google Scholar 

  30. T. Mubeena and P. Sankaran, “Twisted conjugacy classes in lattices in semisimple Lie groups,” Transform. Groups, 19, No. 1, 159–169 (2014).

    Article  MathSciNet  Google Scholar 

  31. T. R. Nasybullov, “Twisted conjugacy classes in general and special linear groups,” Algebra Logika, 51, No. 3, 331–346, 415, 418 (2012).

  32. V. Roman’kov, “Twisted conjugacy classes in nilpotent groups,” J. Pure Appl. Algebra, 215, No. 4, 664–671 (2011).

    Article  MathSciNet  Google Scholar 

  33. J. Taback and P. Wong, “Twisted conjugacy and quasi-isometry invariance for generalized solvable Baumslag–Solitar groups,” J. London Math. Soc. (2), 75, No. 3, 705–717 (2007).

  34. E. Troitsky, “Noncommutative Riesz theorem and weak Burnside type theorem on twisted conjugacy,” Funct. Anal. Appl., 40, No. 2, 117–125 (2006).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    E. V. Troitsky

Authors
  1. E. V. Troitsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. V. Troitsky.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 5, pp. 219–227, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Troitsky, E.V. Two Examples Related to the Twisted Burnside–Frobenius Theory for Infinitely Generated Groups. J Math Sci 248, 661–666 (2020). https://doi.org/10.1007/s10958-020-04903-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-04903-0

Search

Navigation