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

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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.


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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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