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Mathematische Annalen

, Volume 354, Issue 2, pp 765–785 | Cite as

Growth of Schreier graphs of automaton groups

  • Ievgen V. Bondarenko
Article

Abstract

Every automaton group naturally acts on the space X ω of infinite sequences over some alphabet X. For every \({w \in X^{\omega}}\) we consider the Schreier graph Γ w of the action of the group on the orbit of w. We prove that for a large class of automaton groups all Schreier graphs Γ w have subexponential growth bounded above by \({n^{(\log n)^m}}\) with some constant m. In particular, this holds for all groups generated by automata with polynomial activity growth (in terms of S. Sidki), confirming a conjecture of V. Nekrashevych. We present applications to ω-periodic graphs and Hanoi graphs.

Mathematics Subject Classification (2000)

20F65 05C25 20F69 

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References

  1. 1.
    Amir, G., Angel, O., Virag, B.: Amenability of linear-activity automaton groups. J. Eur. Math. Soc. (2009, submitted)Google Scholar
  2. 2.
    Bartholdi L., Grigorchuk R.: On the spectrum of Hecke type operators related to some fractal groups. Proc. Steklov Inst. Math. 231, 1–41 (2000)MathSciNetGoogle Scholar
  3. 3.
    Bartholdi, L., Grigorchuk, R., Nekrashevych, V.: From fractal groups to fractal sets. In: Fractals in Graz 2001, Trends Math., pp. 25–118. Birkhäuser, Basel (2003)Google Scholar
  4. 4.
    Bartholdi L., Kaimanovich V., Nekrashevych V.: On amenability of automata groups. Duke Math. J. 154(3), 575–598 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Benjamini I., Hoffman C.: ω-periodic graphs. Electr. J. Comb. 12, R46 (2005)MathSciNetGoogle Scholar
  6. 6.
    Bondarenko, I.: Groups generated by bounded automata and their Schreier graphs. PhD Dissertation, Texas A&M University (2007)Google Scholar
  7. 7.
    Bondarenko, I., Ceccherini-Silberstein, T., Donno, A., Nekrashevych, V.: On a family of Schreier graphs of intermediate growth associated with a self-similar group. Eur. J. Comb. (2011, accepted)Google Scholar
  8. 8.
    Bondarenko I., Nekrashevych V.: Post-critically finite self-similar groups. Alg. Discrete Math. 2(4), 21–32 (2003)MathSciNetGoogle Scholar
  9. 9.
    Chen X., Shen J.: On the Frame-Stewart conjecture about the Towers of Hanoi. SIAM J. Comput. 33(3), 584–589 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    D’Angeli D., Donno A., Matter M., Nagnibeda T.: Schreier graphs of the Basilica group. J. Mod. Dyn. 4(1), 167–205 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grigorchuk R.: On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR 271(1), 30–33 (1983)MathSciNetGoogle Scholar
  12. 12.
    Grigorchuk R., Nekrashevych V.: Amenable actions of nonabenable groups. Zapiski Nauchnyh Seminarov POMI 326, 85–95 (2005)MATHGoogle Scholar
  13. 13.
    Grigorchuk R., Šuniḱ Z.: Asymptotic aspects of Schreier graphs and Hanoi Towers groups. C. R. Math. Acad. Sci. Paris. 342(8), 545–550 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Grigorchuk R., Šuniḱ Z.: Schreier spectrum of the Hanoi Towers group on three pegs. Proc. Symp. Pure Math. 77, 183–198 (2008)Google Scholar
  15. 15.
    Hinz A.M.: The Tower of Hanoi. Enseign. Math. (2) 35(3–4), 289–321 (1989)MathSciNetMATHGoogle Scholar
  16. 16.
    Klavzar S., Milutinovic U., Petr C.: On the Frame-Stewart algorithm for the multi-peg tower of Hanoi problem. Discrete Appl. Math. 120(1–3), 141–157 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nekrashevych, V.: Self-similar groups. In: Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005)Google Scholar
  18. 18.
    Nekrashevych V.: Free subgroups in groups acting on rooted trees. Groups Geom. Dyn. 4(4), 847–862 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sidki S.: Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity. J. Math. Sci. (New York) 100(1), 1925–1943 (2000)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Sidki S.: Finite automata of polynomial growth do not generate a free group. Geom. Dedicata 108, 193–204 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mechanics and Mathematics DepartmentNational Taras Shevchenko University of KyivKyivUkraine

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