## Institute for Mathematical Sciences

## Preprint ims98-2

** V. Kaimanovich**
* The Poisson Formula for Groups with Hyperbolic Properties.*

Abstract: The Poisson boundary of a group $G$ with a probability measure $\mu$ on it 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, polycyclic groups, some wreath and
semi-direct products including Baumslag--Solitar groups.

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