The Poisson formula for groups with hyperbolic properties

• 成果类型: Article
• 署名作者: Kaimanovich, VA
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2661351
• 发表日期: 2000
• 页码: 659-692
• 关键词: contraction properties; RANDOM MATRICES; random-walks; BOUNDARY; product; SPACES

摘要

The Poisson boundary of a group G with a probability measure mu is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded mu -harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with infinitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semi-simple Lie groups.