Poisson-Furstenberg boundary and growth of groups

• 成果类型: Article
• 署名作者: Bartholdi, Laurent; Erschler, Anna
• 署名单位: Universite PSL; Ecole Normale Superieure (ENS); University of Gottingen
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0712-6
• 发表日期: 2017
• 页码: 347-372
• 关键词: random-walks; amenability; entropy; drift

摘要

We study the Poisson-Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this criterion is useful to establish exponential word growth of groups. We construct groups of exponential growth such that all finitely supported (not necessarily symmetric, possibly degenerate) random walks on these groups have trivial boundary. This gives a negative answer to a question of Kaimanovich and Vershik.