Skip to main content
SpringerLink
Log in

Subshifts with slow complexity and simple groups with the Liouville property

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. Results by Matui and Juschenko-Monod have shown that the derived subgroups of topological full groups of minimal subshifts provide the first examples of finitely generated, simple amenable groups. We show that if the (not necessarily minimal) subshift has a complexity function that grows slowly enough (e.g. linearly), then every symmetric and finitely supported probability measure on the topological full group has trivial Poisson–Furstenberg boundary. We also get explicit upper bounds for the growth of Følner sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Amir, O. Angel, N. Matte Bon, and B. Virág. The Liouville Property for Groups Acting on Rooted Trees (2013) (preprint, arXiv:1307.5652).

  2. Amir G., Angel O., Virág B.: Amenability of linear-activity automaton groups. J. Eur. Math. Soc. (JEMS) 3(15), 705–730 (2013)

    Article  Google Scholar 

  3. Avez A.: Théorème de Choquet-Deny pour les groupes à à croissance non exponentielle. C. R. Acad. Sci. Paris Sér. A, 279, 25–28 (1974)

    MathSciNet  MATH  Google Scholar 

  4. Bartholdi L., Kaimanovich V. A., Nekrashevych V. V.: On amenability of automata groups. Duke Math. J. 3(154), 575–598 (2010)

    Article  MathSciNet  Google Scholar 

  5. Bezuglyi S., Medynets K.: Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems. Colloq. Math. 2(110), 409–429 (2008)

    Article  MathSciNet  Google Scholar 

  6. M. Boyle. Topological orbit equivalence and factor maps in symbolic dynamics. ProQuest LLC, Ann Arbor (1983) (Ph.D. thesis, University of Washington).

  7. Brieussel J.: Amenability and non-uniform growth of some directed automorphism groups of a rooted tree. Math. Z. 2(263), 265–293 (2009)

    Article  MathSciNet  Google Scholar 

  8. Boyle M., Tomiyama J.: Bounded topological orbit equivalence and C*-algebras. J. Math. Soc. Jpn 2(50), 317–329 (1998)

    Article  MathSciNet  Google Scholar 

  9. Bartholdi L., Virág B.: Amenability via random walks. Duke Math. J. 1(130), 39–56 (2005)

    Article  Google Scholar 

  10. Cassaigne J., Karhumäki J.: Toeplitz words, generalized periodicity and periodically iterated morphisms. Eur. J. Combin. 5(18), 497–510 (1997)

    Article  Google Scholar 

  11. J. Cassaigne and F. Nicolas. Factor complexity. In: Combinatorics, automata and number theory, Encyclopedia of Mathematics and Its Applications, Vol. 135. Cambridge University Press, Cambridge (2010), pp. 163–247.

  12. Coulhon Th., Saloff-Coste L.: Minorations pour les chaî nes de Markov unidimensionnelles. Probab. Theory Related Fields 3(97), 423–431 (1993)

    Article  MathSciNet  Google Scholar 

  13. Y. de Cornulier. Groupes pleins-topologiques (d’après Matui, Juschenko, Monod,...) (2013). Written exposition of the Bourbaki Seminar of 19th January 2013. Available at http://www.normalesup.org/~cornulier/.

  14. Y. Derriennic. Quelques applications du théorème ergodique sous-additif. In: Conference on Random Walks (Kleebach, 1979) (French), Astérisque, Vol. 74. Mathematical Society of France, Paris (1980), pp. 183–201.

  15. Erschler A.: Liouville property for groups and manifolds. Invent. Math. 1(155), 55–80 (2004)

    Article  MathSciNet  Google Scholar 

  16. A. Erschler. Poisson–Furstenberg boundaries, large-scale geometry and growth of groups. In: Proceedings of the International Congress of Mathematicians Vol. II. Hindustan Book Agency, New Delhi (2010), pp. 681–704.

  17. Ferenczi S.: Rank and symbolic complexity. Ergodic Theory Dyn. Syst. 4(16), 663–682 (1996)

    MathSciNet  Google Scholar 

  18. N.P. Fogg. Substitutions in dynamics, arithmetics and combinatorics. In: Lecture Notes in Mathematics, Vol. 1794. Springer-Verlag, Berlin (2002) (edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel).

  19. R. Grigorchuk and K. Medynets. On Algebraic Properties of Topological Full Groups (2012) (preprint, arXiv:1105.0719v4).

  20. Gorjuškin A.P.: Imbedding of countable groups in 2-generator simple groups. Mat. Zametki 16, 231–235 (1974)

    MathSciNet  Google Scholar 

  21. Giordano T., Putnam I.F., Skau C.F.: Full groups of Cantor minimal systems. Israel J. Math. 111, 285–320 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Grigorchuk. Milnor’s Problem on the Growth of Groups and its Consequences (2013) (preprint, arXiv:1111.0512).

  23. M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math., (53) (1981), 53–73

  24. P. Hall. On the embedding of a group in a join of given groups. J. Aust. Math. Soc., 17(1974), 434–495 (collection of articles dedicated to the memory of Hanna Neumann, VIII)

  25. Hedlund G.A.: Sturmian minimal sets. Am. J. Math. 66, 605–620 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  26. Higman G.: A finitely generated infinite simple group. J. Lond. Math. Soc. 26, 61–64 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hebisch W., Saloff-Coste L.: Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 2(21), 673–709 (1993)

    Article  MathSciNet  Google Scholar 

  28. K. Juschenko and M. de la Salle. Invariant Means of the Wobbling Group (2013) (preprint, arXiv:1301.4736).

  29. K. Juschenko and N. Monod. Cantor systems, piecewise translations and simple amenable groups. Ann. Math. (2), (2)178 (2013), 775–787

  30. K. Juschenko, V. Nekrashevych, and M. de la Salle. Extensions of Amenable Groups by Recurrent Groupoids (2013) (preprint, arXiv:1305.2637v2).

  31. V. A. Kaimanovich. Münchhausen trick and amenability of self-similar groups. Int. J. Algebra Comput., (5–6)15 (2005), 907–937

  32. Kaĭmanovich V. A., Vershik A. M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 3(11), 457–490 (1983)

    Article  Google Scholar 

  33. M. Lothaire. Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, Vol. 90. Cambridge University Press, Cambridge (2002) (a collective work by Jean Berstel, Dominique Perrin, Patrice Seebold, Julien Cassaigne, Aldo De Luca, Steffano Varricchio, Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Veronique Bruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo, Christophe Reutenauer, Dominique Foata, Guo-Niu Han, Jacques Desarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki and Wojciech Plandowski, With a preface by Berstel and Perrin).

  34. Hiroki Matui. Some remarks on topological full groups of Cantor minimal systems. Int. J. Math., (2)17 (2006), 231–251

  35. Matui H.: Some remarks on topological full groups of Cantor minimal systems II. Ergodic Theory Dyn. Syst. 5(33), 1542–1549 (2013)

    Article  MathSciNet  Google Scholar 

  36. Morse M., Hedlund G. A.: Symbolic dynamics II. Sturmian trajectories. Am. J. Math. 62, 1–42 (1940)

    Article  MathSciNet  Google Scholar 

  37. C. Pittet and L. Saloff-Coste. A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples. In progress, December 2013 version. Available at http://www.math.cornell.edu/~lsc/articles.html.

  38. M. Queffélec. Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, Vol. 1294. Springer-Verlag, Berlin (1987).

  39. Rosenblatt J.: Ergodic and mixing random walks on locally compact groups. Math. Ann. 1(257), 31–42 (1981)

    Article  MathSciNet  Google Scholar 

  40. P.E. Schupp. Embeddings into simple groups. J. Lond. Math. Soc. (2), (1)13 (1976), 90–94

  41. E.K. van Douwen. Measures invariant under actions of F 2. Topol. Appl., (1)34 (1990), 53–68

  42. W. Woess. Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, Vol. 138. Cambridge University Press, Cambridge (2000).

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques et Applications, École Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, France

    Nicolás Matte Bon

  2. Laboratoire de Mathématiques d’Orsay, Université Paris Sud, Faculté des Sciences d’Orsay, 91405, Orsay Cedex, France

    Nicolás Matte Bon

Authors
  1. Nicolás Matte Bon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolás Matte Bon.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matte Bon, N. Subshifts with slow complexity and simple groups with the Liouville property. Geom. Funct. Anal. 24, 1637–1659 (2014). https://doi.org/10.1007/s00039-014-0293-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-014-0293-4

Keywords

Search

Navigation