Israel Journal of Mathematics

, Volume 222, Issue 1, pp 1–19 | Cite as

Some isomorphism results for Thompson-like groups V n (G)

  • Collin BleakEmail author
  • Casey Donoven
  • Julius Jonušas


We find some perhaps surprising isomorphism results for the groups {V n (G)}, where V n (G) is a supergroup of the Higman–Thompson group V n for n ∈ N and GS n , the symmetric group on n points. These groups, introduced by Farley and Hughes, are the groups generated by V n and the tree automorphisms [α] g defined as follows. For each gG and each node α in the infinite rooted n-ary tree, the automorphisms [α] g acts iteratively as g on the child leaves of α and every descendent of α. In particular, we show that V n V n (G) if and only if G is semiregular (acts freely on n points), as well as some additional sufficient conditions for isomorphisms between other members of this family of groups. Essential tools in the above work are a study of the dynamics of the action of elements of V n (G) on the Cantor space, Rubin’s Theorem, and transducers from Grigorchuk, Nekrashevych, and Suschanskiĭ’s rational group on the n-ary alphabet.


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  1. [1]
    L. Bartholdi, V. A. Kaimanovich and V. V. Nekrashevych, On amenability of automata groups, Duke Math. J. 154 (2010), 575–598.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    J. Belk and C. Bleak, Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V, Trans. Amer. Math. Soc. 369 (2017), 3157–3172.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    J. Belk and F. Matucci, Röver’s simple group is of type F , Publ. Mat. 60 (2016), 501–524.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    C. Bleak, P. Cameron, Y. Maissel, A. Navas and F. Olukoya, The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman groups G n,r, Submitted (2015), 1–44.Google Scholar
  5. [5]
    C. Bleak and D. Lanoue, A family of non-isomorphism results, Geom. Dedicata 146 (2010), 21–26.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    C. Bleak, F. Matucci and M. Neunhöffer, Embeddings into Thompson’s group V and coCF groups, J. Lond. Math. Soc. (2) 94 (2016), 583–597.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    C. Bleak and M. Quick, The infinite simple group V of Richard J. Thompson: presentations by permutations, Groups Geom. Dyn., to appear (2017), 1–26.Google Scholar
  8. [8]
    K. S. Brown, Finiteness properties of groups, in Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), Vol. 44, 1987, pp. 45–75.zbMATHMathSciNetGoogle Scholar
  9. [9]
    D. S. Farley, Local similarity groups with context-free co-word problem, ArXiv e-prints (2014), 17.Google Scholar
  10. [10]
    D. S. Farley and B. Hughes, Finiteness properties of some groups of local similarities, Proc. Edinb. Math. Soc. (2) 58 (2015), 379–402.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    V. M. Gluškov, Abstract theory of automata, Uspehi Mat. Nauk 16 (1961), 3–62.MathSciNetGoogle Scholar
  12. [12]
    R. I. Grigorchuk, V. V. Nekrashevich and V. I. Sushchanskiĭ, Automata, dynamical systems, and groups, Tr. Mat. Inst. Steklova 231 (2000), 134–214.zbMATHMathSciNetGoogle Scholar
  13. [13]
    G. Higman, Finitely presented infinite simple groups, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974, Notes on Pure Mathematics, No. 8 (1974).Google Scholar
  14. [14]
    D. A. Huffman, The synthesis of sequential switching circuits. I, II, J. Franklin Inst. 257 (1954), 161–190, 275–303.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    B. Hughes, Local similarities and the Haagerup property (with an appendix by D. S. Farley), Groups Geom. Dyn. 3 (2009), 299–315.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    S. McCleary and M. Rubin, Locally moving groups and the reconstruction problem for chains and circles, ArXiv e-prints (1995), 1–171.Google Scholar
  17. [17]
    V. Nekrashevych, Cuntz–Pimsner algebras of group actions., J. Operator Theory 52 (2004), 223–249.MathSciNetGoogle Scholar
  18. [18]
    C. E. Röver, Constructing finitely presented simple groups that contain Grigorchuk groups, J. Algebra 220 (1999), 284–313.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    M. Rubin, Locally moving groups and reconstruction problems, in Ordered groups and infinite permutation groups, Math. Appl., Vol. 354, Kluwer Acad. Publ., Dordrecht, 1996, pp. 121–157.CrossRefGoogle Scholar
  20. [20]
    E. A. Scott, A construction which can be used to produce finitely presented infinite simple groups, J. Algebra 90 (1984), 294–322.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    E. A. Scott, The embedding of certain linear and abelian groups in finitely presented simple groups, J. Algebra 90 (1984), 323–332.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    E. A. Scott, A finitely presented simple group with unsolvable conjugacy problem, J. Algebra 90 (1984), 333–353.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    P. V. Silva and B. Steinberg, On a class of automata groups generalizing lamplighter groups, Internat. J. Algebra Comput. 15 (2005), 1213–1234.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    W. Thumann, Operad groups and their finiteness properties, Adv. Math. 307 (2017), 417–487.CrossRefzbMATHMathSciNetGoogle Scholar

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsMathematical Institute, North HaughSt Andrews, FifeUK

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