Entropy and drift in word hyperbolic groups

Abstract : The fundamental inequality of Guivarc'h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary hyperbolic group which is not virtually free, endowed with a word distance, the fundamental inequality is strict for symmetric measures with finite support, uniformly for measures with a given support. This answers a conjecture of S. Lalley. For admissible measures, this is proved using previous results of Ancona and Blachère-Haïssinsky-Mathieu. For non-admissible measures, this follows from a counting result, interesting in its own right: we show that, in any infinite index subgroup, the number of non-distorted points is exponentially small. The uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup.
Type de document :
Article dans une revue
Inventiones Mathematicae, Springer Verlag, 2018, 211 (3), pp.1201-1255. 〈10.1007/s00222-018-0788-y〉
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01107467
Contributeur : Sebastien Gouezel <>
Soumis le : mardi 20 janvier 2015 - 17:31:41
Dernière modification le : jeudi 21 juin 2018 - 17:14:02
Document(s) archivé(s) le : mardi 21 avril 2015 - 11:50:35

Fichiers

hlv.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

Citation

Sébastien Gouëzel, Frédéric Mathéus, François Maucourant. Entropy and drift in word hyperbolic groups. Inventiones Mathematicae, Springer Verlag, 2018, 211 (3), pp.1201-1255. 〈10.1007/s00222-018-0788-y〉. 〈hal-01107467〉

Partager

Métriques

Consultations de la notice

456

Téléchargements de fichiers

143