Lattice action on the boundary of SL $(n,\mathbb{R})}$

Abstract

Let $\Gamma$ be a lattice in $G={\rm SL}(n,\mathbb{R})$ 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.