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 be a discrete group, and { a probability measure on G. A function f on G is called-harmonic if f(g) = P f(gx) (x) 8 g 2 G. The Poisson boundary of the pair (G;) is the probability space (?;) with a measure type preserving action of the group G uniquely determined by the following condition: the Poisson formula f(g) = h b f; gi is an isometry between the space H 1 (G;) of bounded-harmonic functions with the sup-norm and the space L 1 (?;). The problem of describing the Poisson boundary in terms of algebraic or geometric objects associated with the group splits into two parts: rst, to nd a space (B;) such that the Poisson formula gives an embedding of L 1 (B;) into H 1 (G;) (such spaces are called-boundaries), and, second, to show that a given-boundary is maximal (i.e., coincides with the whole Poisson boundary) K4]. The methods used for identifying the Poisson boundary in the case of Lie groups (e.g., see F], R]) are unapplicable to discrete groups. The author has suggested a new approach based on the entropy theory of random walks which leads to two simple geometric criteria of boundary maximality (Theorems 1 and 2). These criteria are easily applicable to discrete subgroups of semi-simple Lie groups (as well as in a large number of other situations K3], K5], KM]).