Given a group G and a positive integer d ≥ 2 we introduce the notion of an automaton rank of a group G with respect to its self-similar actions on a d-ary tree of words as the minimal number of states in an automaton over a d-letter alphabet which generates this group (topologically if G is closed). We construct minimal automata generating free abelian groups of finite ranks, which completely determines automaton ranks of free abelian groups. We also provide naturally defined 3-state automaton realizations for profinite groups which are infinite wreath powers … ≀ H ≀ H for some 2-generated finite perfect groups H. This determines the topological rank and improves the estimation for the automaton rank of these wreath powers. We show that we may take H as alternating groups and projective special linear groups.


Tree of Words Self-similar Group Automaton Group Wreath Product 


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  1. 1.
    Bartholdi, L., Sunik, Z.: Some solvable automaton groups. Contemporary Mathematics 394, 11–29 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bhattacharjee, M.: The probability of generating certain profinite groups by two elemets. Israel Journal of Mathematics 86, 311–329 (1994)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bondarenko, I.: Finite generation of iterated wreath products. Archiv der Mathematik 95(4), 301–308 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bondarenko, I., Grigorchuk, R., Kravchenko, R., Muntyan, Y., Nekrashevych, V., Sunic, Z., Savchuk, D.: On classification of groups generated by 3-state automata over a 2-letter alphabet. Algebra and Discrete Mathematics 1, 1–163 (2008)MathSciNetMATHGoogle Scholar
  5. 5.
    Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V., Paterson, M.S., Thurston, W.P.: Word processing in groups. Jones and Bartlett Publishers, Boston (1992)MATHGoogle Scholar
  6. 6.
    Grigorchuk, R., Nekrashevych, V., Sushchanskyy, V.: Automata, dynamical systems and groups. Proceedings of Steklov Institute of Mathematics 231, 128–203 (2000)MathSciNetGoogle Scholar
  7. 7.
    Grigorchuk, R., Zuk, A.: The lamplighter group as a group generated by a 2-state automaton and its spectrum. Geometriae Dedicata 87, 209–244 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kharlampovich, O., Khoussainov, B., Miasnikov, A.: From automatic structures to automatic groups, 35 pages (2011), arXiv:1107.3645v2 [math.GR] Google Scholar
  9. 9.
    Khoussainov, B., Nerode, A.: Automatic Presentations of Structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  10. 10.
    Nekrashevych, V.: Self-similar groups. Mathematical Surveys and Monographs, vol. 117. Amer. Math. Soc., Providence (2005)MATHGoogle Scholar
  11. 11.
    Nekrashevych, V., Sidki, S.: Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms. In: Groups: Topological, Combinatorial and Arithmetic Aspects. LMS Lecture Notes Series, vol. 311, pp. 375–404 (2004)Google Scholar
  12. 12.
    Quick, M.: Probabilistic generation of wreath products of non-abelian finite simple groups. Commun. Algebra 32(12), 4753–4768 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Silva, P., Steinberg, B.: On a class of automata groups generalizing lamplighter groups. Internat. J. Algebra Comput. 15(5-6), 1213–1234 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Steinberg, B., Vorobets, M., Vorobets, Y.: Automata over a binary alphabet generating free groups of even rank. Internat. J. Algebra Comput. 21(1-2), 329–354 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Vorobets, M., Vorobets, Y.: On a series of finite automata defining free transformation groups. Groups Geom. Dyn. 4(2), 377–405 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Adam Woryna
    • 1
  1. 1.Institute of MathematicsSilesian University of TechnologyGliwicePoland

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