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

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.

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References

  1. 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. 2.

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

  3. 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. 4.

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

  5. 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).

    MathSciNet  Article  Google Scholar 

  6. 6.

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

  7. 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. 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. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 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. 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. 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).

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 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. 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. 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. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 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).

    MathSciNet  Article  Google Scholar 

  19. 19.

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

  20. 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. 21.

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

    MathSciNet  Article  Google Scholar 

  22. 22.

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

    MathSciNet  Article  Google Scholar 

  23. 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. 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).

    MathSciNet  Article  Google Scholar 

  25. 25.

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

    MathSciNet  Article  Google Scholar 

  26. 26.

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

  27. 27.

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

  28. 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).

    MathSciNet  Article  Google Scholar 

  29. 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).

    MathSciNet  Article  Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

  31. 31.

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

  32. 32.

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

    MathSciNet  Article  Google Scholar 

  33. 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. 34.

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

    MathSciNet  Article  Google Scholar 

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Correspondence to E. V. Troitsky.

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

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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

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