Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The Liouville property for groups acting on rooted trees

Gideon Amir, Omer Angel, Nicolás Matte Bon, and Bálint Virág

Full-text: Open access

Enhanced PDF (313 KB)

Abstract

We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure.

Résumé

Nous démontrons que les groupes engendrés par les automates d’activité bornée ont la propriété de Liouville pour tout choix d’une mesure de probabilité symétrique, de support fini. Plus généralement, nous montrons ce résultat pour tous les groupes agissants sur un arbre enraciné par automorphismes de type borné. Dans le cas des groupes d’automate nous obtenons aussi une borne supérieure uniforme pour l’entropie, qui ne dépend pas du choix de la mesure symétrique, de support fini.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1763-1783.

Dates
Received: 16 October 2013
Revised: 2 November 2014
Accepted: 7 July 2015
First available in Project Euclid: 17 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1479373247

Digital Object Identifier
doi:10.1214/15-AIHP697

Mathematical Reviews number (MathSciNet)
MR3573294

Zentralblatt MATH identifier
06673647

Subjects
Primary: 20F69: Asymptotic properties of groups 05C81: Random walks on graphs 20E08: Groups acting on trees [See also 20F65]

Keywords
Groups acting on rooted trees Liouville property Random walk entropy Recurrent Schreier graphs

Citation

Amir, Gideon; Angel, Omer; Matte Bon, Nicolás; Virág, Bálint. The Liouville property for groups acting on rooted trees. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1763--1783. doi:10.1214/15-AIHP697. https://projecteuclid.org/euclid.aihp/1479373247


Export citation

References

  • [1] G. Amir, O. Angel and B. Virág. Amenability of linear-activity automaton groups. J. Eur. Math. Soc. (JEMS) 15 (3) (2013) 705–730.
    Mathematical Reviews (MathSciNet): MR3085088
    Zentralblatt MATH: 1277.37019
    Digital Object Identifier: doi:10.4171/JEMS/373
  • [2] G. Amir and B. Virág. Speed exponents for random walks on groups. Preprint, 2012. Available at arXiv:1203.6226.
    arXiv: 1203.6226
  • [3] G. Amir and B. Virág. Positive speed for high-degree automaton groups. Groups Geom. Dyn. 8 (1) (2014) 23–38.
    Mathematical Reviews (MathSciNet): MR3209701
    Zentralblatt MATH: 1294.05142
    Digital Object Identifier: doi:10.4171/GGD/215
  • [4] L. Bartholdi and A. Erschler. Poisson–Furstenberg boundary and growth of groups. Preprint, 2011. Available at arXiv:1107.5499.
    arXiv: 1107.5499
  • [5] L. Bartholdi, V. A. Kaimanovich and V. V. Nekrashevych. On amenability of automata groups. Duke Math. J. 154 (3) (2010) 575–598.
    Mathematical Reviews (MathSciNet): MR2730578
    Zentralblatt MATH: 1185.37020
    Digital Object Identifier: doi:10.1215/00127094-2010-046
    Project Euclid: euclid.dmj/1283865313
  • [6] L. Bartholdi and B. Virág. Amenability via random walks. Duke Math. J. 130 (1) (2005) 39–56.
    Mathematical Reviews (MathSciNet): MR2176547
    Zentralblatt MATH: 1104.43002
    Digital Object Identifier: doi:10.1215/S0012-7094-05-13012-5
    Project Euclid: euclid.dmj/1131804019
  • [7] I. Bondarenko. Groups Generated by Bounded Automata and Their Schreier Graphs. Thesis (Ph.D.)–Texas, A&M University. ProQuest LLC, Ann Arbor, MI, 2007.
    Mathematical Reviews (MathSciNet): MR2711289
  • [8] J. Brieussel. Amenability and non-uniform growth of some directed automorphism groups of a rooted tree. Math. Z. 263 (2) (2009) 265–293.
    Mathematical Reviews (MathSciNet): MR2534118
    Zentralblatt MATH: 1227.43001
    Digital Object Identifier: doi:10.1007/s00209-008-0417-3
  • [9] J. Brieussel. Behaviors of entropy on finitely generated groups. Ann. Probab. 41 (6) (2013) 4116–4161.
    Mathematical Reviews (MathSciNet): MR3161471
    Zentralblatt MATH: 1280.05123
    Digital Object Identifier: doi:10.1214/12-AOP761
    Project Euclid: euclid.aop/1384957784
  • [10] J. Brieussel. Følner sets of alternate directed groups. Ann. Inst. Fourier (Grenoble) 64 (3) (2014) 1109–1130.
    Mathematical Reviews (MathSciNet): MR3330165
    Zentralblatt MATH: 1310.43001
    Digital Object Identifier: doi:10.5802/aif.2875
  • [11] Y. Derriennic. Quelques applications du théorème ergodique sous-additif. In Conference on Random Walks (Kleebach, 1979) (French) 183–201, 4. Astérisque 74. Soc. Math. France, Paris, 1980.
    Mathematical Reviews (MathSciNet): MR588163
  • [12] A. Erschler. Boundary behavior for groups of subexponential growth. Ann. of Math. (2) 160 (3) (2004) 1183–1210.
    Mathematical Reviews (MathSciNet): MR2144977
    Zentralblatt MATH: 1089.20025
    Digital Object Identifier: doi:10.4007/annals.2004.160.1183
  • [13] A. Erschler. Liouville property for groups and manifolds. Invent. Math. 155 (1) (2004) 55–80.
    Mathematical Reviews (MathSciNet): MR2025301
    Zentralblatt MATH: 1043.60006
    Digital Object Identifier: doi:10.1007/s00222-003-0314-7
  • [14] R. Grigorchuk and Z. Šuniḱ. Asymptotic aspects of Schreier graphs and Hanoi Towers groups. C. R. Math. Acad. Sci. Paris 342 (8) (2006) 545–550.
    Mathematical Reviews (MathSciNet): MR2217913
    Zentralblatt MATH: 1135.20016
    Digital Object Identifier: doi:10.1016/j.crma.2006.02.001
  • [15] R. I. Grigorchuk. On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR 271 (1) (1983) 30–33.
    Mathematical Reviews (MathSciNet): MR712546
    Zentralblatt MATH: 0547.20025
  • [16] R. I. Grigorchuk and A. Żuk. On a torsion-free weakly branch group defined by a three state automaton. Internat. J. Algebra Comput. 12 (1–2) (2002) 223–246. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000).
    Mathematical Reviews (MathSciNet): MR1902367
    Zentralblatt MATH: 1070.20031
    Digital Object Identifier: doi:10.1142/S0218196702001000
  • [17] N. Gupta and S. Sidki. On the Burnside problem for periodic groups. Math. Z. 182 (3) (1983) 385–388.
    Mathematical Reviews (MathSciNet): MR696534
    Zentralblatt MATH: 0513.20024
    Digital Object Identifier: doi:10.1007/BF01179757
  • [18] K. Juschenko, V. Nekrashevych and M. de la Salle. Extensions of amenable groups by recurrent groupoids. Preprint, 2013. Available at arXiv:1305.2637v1.
    arXiv: 1305.2637v1
  • [19] V. A. Kaimanovich. “Münchhausen trick” and amenability of self-similar groups. Internat. J. Algebra Comput. 15 (5–6) (2005) 907–937.
    Mathematical Reviews (MathSciNet): MR2197814
    Digital Object Identifier: doi:10.1142/S0218196705002694
  • [20] V. A. Kaĭmanovich and A. M. Vershik. Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11 (3) (1983) 457–490.
    Mathematical Reviews (MathSciNet): MR704539
    Digital Object Identifier: doi:10.1214/aop/1176993497
    Project Euclid: euclid.aop/1176993497
  • [21] A. Karlsson and F. Ledrappier. Linear drift and Poisson boundary for random walks. Pure Appl. Math. Q. 3 (4, Special Issue: In honor of Grigory Margulis. Part 1) (2007) 1027–1036.
    Mathematical Reviews (MathSciNet): MR2402595
    Digital Object Identifier: doi:10.4310/PAMQ.2007.v3.n4.a8
  • [22] R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge Univ. Press, Cambridge, 2015. Available at http://pages.iu.edu/~rdlyons/.
  • [23] V. Nekrashevych. Self-Similar Groups. Mathematical Surveys and Monographs 117. American Mathematical Society, Providence, RI, 2005.
    Mathematical Reviews (MathSciNet): MR2162164
    Zentralblatt MATH: 1087.20032
  • [24] V. Nekrashevych. Free subgroups in groups acting on rooted trees. Groups Geom. Dyn. 4 (4) (2010) 847–862.
    Mathematical Reviews (MathSciNet): MR2727668
    Zentralblatt MATH: 1267.37009
    Digital Object Identifier: doi:10.4171/GGD/110
  • [25] V. Nekrashevych. Iterated monodromy groups. In Groups St Andrews 2009 in Bath. Vol. 1 41–93. London Math. Soc. Lecture Note Ser. 387. Cambridge Univ. Press, Cambridge, 2011.
    Mathematical Reviews (MathSciNet): MR2858850
    Zentralblatt MATH: 1235.37016
    Digital Object Identifier: doi:10.1017/CBO9780511842467.004
  • [26] J. Rosenblatt. Ergodic and mixing random walks on locally compact groups. Math. Ann. 257 (1) (1981) 31–42.
    Mathematical Reviews (MathSciNet): MR630645
    Zentralblatt MATH: 0451.60011
    Digital Object Identifier: doi:10.1007/BF01450653
  • [27] S. Sidki. Finite automata of polynomial growth do not generate a free group. Geom. Dedicata 108 (2004) 193–204.
    Mathematical Reviews (MathSciNet): MR2112674
    Zentralblatt MATH: 1075.20011
    Digital Object Identifier: doi:10.1007/s10711-004-2368-0

Institut Henri Poincaré

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more