Russian Journal of Mathematical Physics

, Volume 26, Issue 1, pp 122–129 | Cite as

Reidemeister Classes in Some Weakly Branch Groups

  • E. V. TroitskyEmail author


We prove that a saturated weakly branch group G on an infinite spherically symmetric rooted tree T (i.e., a group which acts on T faithfully, level-transitively, with nontrivial rigid stabilizers of all vertices, and with a transitive action on sub-trees of some characteristic subgroups of all level stabilizers) has the property R (any automorphism ϕ: GG has infinite Reidemeister number) in each of the following cases: (1) any element of Out(G) is of finite order; (2) for any ϕ, the number of orbits on levels of the tree automorphism t, such that ϕ(g) = tgt−1, is uniformly bounded and G is weakly stabilizer transitive, i.e., the intersection of stabilizers of all vertices of any level, except for successors of one vertex of the previous level, acts transitively on these successors; (3) G is finitely generated, has prime branching numbers, and is weakly stabilizer transitive with some non-Abelian quotients of stabilizers (with no restrictions on automorphisms). Some related facts and generalizations are proved.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. Dekimpe and D. Gonçalves, “The R∞ Property for Free Groups, Free Nilpotent Groups and Free Solvable Groups,” Bull. Lond. Math. Soc. 46 (4), 737–746 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. D. Dixon and B. Mortimer, Permutation Groups, Vol. 163 Graduate Texts in Mathematics (Springer–Verlag, New York, 1996).Google Scholar
  3. 3.
    A. Fel’shtyn, Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion 699, Mem. Amer. Math. Soc. (AMS, Providence, R.I., 2000).Google Scholar
  4. 4.
    A. Fel’shtyn and R. Hill, “The Reidemeister Zeta Function with Applications to Nielsen Theory and a Connection with Reidemeister Torsion,” K–Theory 8 (4), 367–393 (1994).MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Fel’shtyn, Y. Leonov, and E. Troitsky, “Twisted Conjugacy Classes in Saturated Weakly Branch Groups,” Geom. Dedicata 134, 61–73 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Fel’shtyn, N. Luchnikov, and E. Troitsky, “Twisted Inner Representations,” Russ. J. Math. Phys. 22 (3), 301–306 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Fel’shtyn and T. Nasybullov, “The R and S Properties for Linear Algebraic Groups,” J. Group Theory 19 (5), 901–921 (2016).MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. Fel’shtyn and E. Troitsky, “Twisted Burnside–Frobenius Theory for Discrete Groups,” J. Reine Angew. Math. 613, 193–210 (2007).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. Fel’shtyn and E. Troitsky, “Aspects of the Property R ,” J. Group Theory 18 (6), 1021–1034 (2015).MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. Fel’shtyn and E. Troitsky, “Twisted Burnside–Frobenius Theory for Endomorphisms of Polycyclic Groups,” Russ. J. Math. Phys. 25 (1), 17–26 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Fel’shtyn, E. Troitsky, and A. Vershik, “Twisted Burnside Theorem for Type II1 Groups: an Example,” Math. Res. Lett. 13 (5), 719–728 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    D. Gonçalves and P. Wong, “Twisted Conjugacy Classes in Nilpotent Groups,” J. Reine Angew. Math. 633, 11–27 (2009).MathSciNetzbMATHGoogle Scholar
  13. 13.
    R. I. Grigorchuk, “On Burnside’s Problem on Periodic Groups,” Funct. Anal. Appl. 14, 41–43 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    R. I. Grigorchuk and S. N. Sidki, “The Group of Automorphisms of a 3–Generated 2–Group of Intermediate Growth,” Internat. J. Algebra Comput. 14 (5–6), 667–676 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    N. Gupta and S. Sidki, “On the Burnside Problem for Periodic Groups,” Math. Z. 182, 385–388 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    E. Jabara, “Automorphisms with Finite Reidemeister Number in Residually Finite Groups,” J. Algebra 320 (10), 3671–3679 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Y. Lavreniuk and V. Nekrashevych, “Rigidity of Branch Groups Acting on Rooted Trees,” Geom. Dedicata 89, 159–179 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    T. Mubeena and P. Sankaran, “Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups,” Canad. Math. Bull. 57 (1), 132–140 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    V. Roman’kov, “Twisted Conjugacy Classes in Nilpotent Groups,” J. Pure Appl. Algebra 215 (4), 664–671 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    P. Rowley, “Finite Groups Admitting a Fixed–Point–Free Automorphism Group,” J. Algebra 174 (2), 724–727 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    S. Sidki, “On a 2–Generated Infinite 3–Group: Subgroups and Automorphisms,” J. Algebra 110 (1), 24–55 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J. Taback and P. Wong, “Twisted Conjugacy and Quasi–Isometry Invariance for Generalized Solvable Baumslag–Solitar Groups,” J. Lond. Math. Soc. (2) 75 (3), 705–717 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    E. Troitsky, “Reidemeister Classes in Lamplighter Type Groups,” E–print, arXiv:1711.09371, 2017 (accepted in Communications in Algebra, Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Dept. of Mech. and Math.Moscow State UniversityGSP-1 MoscowRussia

Personalised recommendations