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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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


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