[Paper Review] The Poisson formula for groups with hyperbolic properties

Vadim A. Kaimanovich|ArXiv.org|Feb 15, 1998

advanced mathematical theories4 citations

TL;DR

This paper introduces a novel geometric method for identifying the Poisson boundary of groups with hyperbolic properties using entropy estimates for conditional random walks. It establishes simple criteria for boundary maximality, enabling the identification of the Poisson boundary with natural topological boundaries in word hyperbolic groups, discontinuous isometry groups of Gromov hyperbolic spaces, groups with infinitely many ends, cocompact lattices in Cartan-Hadamard manifolds, and discrete subgroups of semisimple Lie groups.

ABSTRACT

The Poisson boundary of a group G with a probability measure μis 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 μ-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.

Motivation & Objective

• To develop a geometric criterion for identifying the Poisson boundary of groups with hyperbolic-like properties.
• To provide a unified approach to boundary identification across diverse classes of groups, including word hyperbolic and semisimple Lie group subgroups.
• To establish conditions under which the Poisson boundary is maximal, i.e., captures all bounded harmonic functions.
• To connect probabilistic harmonic analysis with geometric group theory via entropy-based methods.
• To extend the classical Poisson formula to groups with hyperbolic structure using intrinsic geometric invariants.

Proposed method

• Utilizes entropy estimates for conditional random walks as the central analytical tool.
• Applies geometric properties of Gromov hyperbolic spaces and their isometry groups to derive boundary criteria.
• Employs the concept of ergodic components of the time shift on path space to define the Poisson boundary.
• Introduces a criterion based on asymptotic entropy to determine when the Poisson boundary is maximal.
• Relies on the structure of random walks on groups with hyperbolic-like geometry to link entropy decay to boundary identification.
• Uses the interplay between measure-theoretic entropy and geometric boundary structure to characterize the Poisson boundary.

Experimental results

Research questions

• RQ1 Under what geometric conditions is the Poisson boundary of a group maximal?
• RQ2 How can entropy estimates for conditional random walks be used to identify the Poisson boundary?
• RQ3 In which classes of groups with hyperbolic properties can the Poisson boundary be identified with a natural topological boundary?
• RQ4 What is the relationship between the geometric structure of a group and the structure of its harmonic functions?
• RQ5 Can a single geometric criterion unify the identification of the Poisson boundary across diverse hyperbolic-like groups?

Key findings

• The Poisson boundary of word hyperbolic groups is identified with their Gromov boundary via the entropy-based criterion.
• For discontinuous groups of isometries of Gromov hyperbolic spaces, the Poisson boundary coincides with the Gromov boundary under the entropy maximality condition.
• Groups with infinitely many ends have their Poisson boundary identified with the space of ends, as shown by entropy analysis.
• Cocompact lattices in Cartan-Hadamard manifolds have their Poisson boundary identified with the ideal boundary of the symmetric space.
• Discrete subgroups of semisimple Lie groups with finite covolume have their Poisson boundary identified with the Furstenberg boundary under the entropy criterion.
• The method provides a unified framework that establishes the maximality of the Poisson boundary across all these classes using geometric and entropy-based arguments.