The Poisson formula for groups with hyperbolic properties

@article{Kaimanovich1998ThePF,
  title={The Poisson formula for groups with hyperbolic properties},
  author={Vadim A. Kaimanovich},
  journal={Annals of Mathematics},
  year={1998},
  volume={152},
  pages={659-692}
}
  • V. Kaimanovich
  • Published 15 February 1998
  • Mathematics
  • Annals of Mathematics
The Poisson boundary of a group G with a probability measure „ 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 „-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… 
POISSON BOUNDARY OF GROUPS ACTING ON R-TREES
We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic
An approach to traces of random walks on the boundary of a hyperbolic group via reflected Dirichlet spaces
On a non elementary, Gromov hyperbolic group of conformal dimension less than two - say a surface group - we consider a symmetric probability measure whose support generates the whole group and with
Boundary and Entropy of Space Homogeneous Markov Chains
We study the Poisson boundary (≡ representation of bounded harmonic functions) of Markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations.
Poisson boundary of groups acting on ℝ-trees
We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic
Harmonic Functions on Discrete Subgroups of Semi-simple Lie Groups
A description of the Poisson boundary of random walks on discrete subgroups of semi-simple Lie groups in terms of geometric boundaries of the corresponding Riemannian symmetric spaces is given. Let G
Poisson boundary of groups acting on real trees
We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic
Continuity of asymptotic characteristics for random walks on hyperbolic groups
We describe a new approach to proving the continuity of asymptotic entropy as a function of a transition measure under a finite first moment condition. It is based on using conditional random walks
Dimensional properties of the harmonic measure for a random walk on a hyperbolic group
This paper deals with random walks on isometry groups of Gromov hyperbolic spaces, and more precisely with the dimension of the harmonic measure $\nu$ associated with such a random walk. We first
Discrete random walks on the group Sol
The harmonic measure ν on the boundary of the group Sol associated to a discrete random walk of law µ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with
The Poisson boundary of CAT(0) cube complex groups
We consider a finite-dimensional, locally finite CAT(0) cube complex X admitting a co-compact properly discontinuous countable group of automorphisms G. We construct a natural compact metric space