  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.
