Duke Mathematical Journal

On amenability of automata groups

Laurent Bartholdi, Vadim A. Kaimanovich, and Volodymyr V. Nekrashevych

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We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (mother groups) which is done by analyzing the asymptotic properties of random walks on these groups.

Article information

Duke Math. J. Volume 154, Number 3 (2010), 575-598.

First available in Project Euclid: 7 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 20P05: Probabilistic methods in group theory [See also 60Bxx] 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 43A07: Means on groups, semigroups, etc.; amenable groups 60G50: Sums of independent random variables; random walks


Bartholdi, Laurent; Kaimanovich, Vadim A.; Nekrashevych, Volodymyr V. On amenability of automata groups. Duke Math. J. 154 (2010), no. 3, 575--598. doi:10.1215/00127094-2010-046. http://projecteuclid.org/euclid.dmj/1283865313.

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  • G. Amir, O. Angel, and B. Virág, Amenability of linear-activity automaton groups, preprint.
  • A. Avez, Entropie des groupes de type fini, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1363--A1366.
  • L. Bartholdi, A Wilson group of non-uniformly exponential growth, C. R. Math. Acad. Sci. Paris 336 (2003), 549--554.
  • L. Bartholdi and R. I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Proc. Steklov Inst. Math. 2000, no. 4, 1--41.
  • L. Bartholdi and B. Virág, Amenability via random walks, Duke Math. J. 130 (2005), 39--56.
  • G. Baumslag, Topics in Combinatorial Group Theory, Lectures Math. ETH Zürich, Birkhäuser, Basel, 1993.
  • E. Bondarenko and V. Nekrashevych, Post-critically finite self-similar groups, Algebra Discrete Math. 2003, 21--32.
  • A. M. Brunner, S. Sidki, and A. C. Vieira, A just nonsolvable torsion-free group defined on the binary tree, J. Algebra 211 (1999), 99--114.
  • T. Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal. 141 (1996), 510--539.
  • E. F. Da Silva, Uma famí lia de grupos quase não-solúveis definida sobre árvores $n$-árias, $n\ge 2$, Ph.D. dissertation, Universidade de Brasilia, 2001.
  • M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509--544.
  • P. De Lya Arp, R. I. Grigorchuk, and T. Chekerini-Sil'Berstaĭn, Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces, Proc. Steklov Inst. Math. 224 (1999), 57--97.
  • A. Erschler, On isoperimetric profiles of finitely generated groups, Geom. Dedicata 100 (2003), 157--171.
  • —, Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks, Probab. Theory Related Fields 136 (2006), 560--586.
  • J. Fabrykowski and N. Gupta, On groups with sub-exponential growth functions, II, J. Indian Math. Soc. (N.S.) 56 (1991), 217--228.
  • R. I. Grigorchuk, On Burnside's problem on periodic groups, Functional. Anal. Appl. 14 (1980), 41--43.
  • —, Degrees of growth of finitely generated groups and the theory of invariant means, Math. USSR-Izv. 25 (1985), no. 2, 259--300.
  • —, An example of a finitely presented amenable group that does not belong to the class EG, Sb. Math. 189 (1998), no. 1-2, 75--95.
  • R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskiĭ, Automata, dynamical systems, and groups, Proc. Steklov. Inst. Math. 2000, no. 4, 128--203.
  • R. I. Grigorchuk and A. \DZuk, ``On a torsion-free weakly branch group defined by a three state automaton'' in International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, Neb., 2000), Internat. J. Algebra Comput. 12, 2002, 223--246.
  • A. Grigor'Yan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana 10 (1994), 395--452.
  • N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), 385--388.
  • V. A. Kaimanovich, ``Münchhausen trick'' and amenability of self-similar groups, Internat. J. Algebra Comput. 15 (2005), 907--937.
  • V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: Boundary and entropy, Ann. Probab. 11 (1983), 457--490.
  • V. Nekrashevych, Self-Similar Groups, Math. Surveys Monogr. 117, Amer. Math. Soc., Providence, 2005.
  • P. M. Neumann, Some questions of Edjvet and Pride about infinite groups, Illinois J. Math. 30 (1986), 301--316.
  • C. Pittet and L. Saloff-Coste, ``Amenable groups, isoperimetric profiles and random walks'' in Geometric Group Theory Down Under (Canberra, 1996), de Gruyter, Berlin, 1999, 293--316.
  • —, On the stability of the behavior of random walks on groups, J. Geom. Anal. 10 (2000), 713--737.
  • V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk 22 (1967), no. 5, 3--56.
  • D. Segal, The finite images of finitely generated groups, Proc. London Math. Soc. (3) 82 (2001), 597--613.
  • S. Sidki, ``Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity'' in Algebra, 12, J. Math. Sci. (New York) 100, Kluwer, New York, 2000, 1925--1943.
  • —, Finite automata of polynomial growth do not generate a free group, Geom. Dedicata 108 (2004), 193--204.
  • J. Von Neumann, Zur allgemeinen Theorie des Maßes, Fundamenta 13 (1929), 73--116.
  • J. S. Wilson, On exponential growth and uniformly exponential growth for groups, Invent. Math. 155 (2004), 287--303.
  • W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Math. 138, Cambridge Univ. Press, Cambridge, 2000.