Lattice action on the boundary of SL(n, R)

Let Gamma be a lattice in G = SL(n, R) and X = G/S be a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish the uniform distribution of orbits of Gamma in X analogous to the classical equidistribution on a torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of a Borel subgroup of G.