Entropy and drift in word hyperbolic groups

• 成果类型: Article
• 署名作者: Gouezel, Sebastien; Matheus, Frederic; Maucourant, Francois
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Nantes Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Rennes
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0788-y
• 发表日期: 2018
• 页码: 1201-1255
• 关键词: quasi-conformal measures; random-walks; harmonic measure; BOUNDARY; spectrum

摘要

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 BlachSre-Ha < ssinsky-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 compared to the growth of balls in the whole group. The uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup.