This paper is devoted to the
study of harmonic functions on groups. The approach is via a detailed study of the
Poisson flow associated with a Borel probability measure μ on a locally compact
group T. Again the basic idea is that though many results associated with the
study of harmonic functions on groups are couched in probabilistic terms
and proved using methods of probability theory, they really belong in the
domain of topological dynamics. The major results include a proof that a
solvable connected Lie group admits only constants as harmonic functions for a
spread out measure μ with μ(A) = μ(A−1) for all Borel sets A, and a new
non-geometric proof of a fundamental result of Furstenberg’s on semi-simple Lie
groups.