Orbits of discrete subgroups on a symmetric space and the Furstenberg   boundary
  Let X be a symmetric space of noncompact type and \Gamma a lattice in the isometry group of X. We study the distribution of orbits of \Gamma acting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in X and b in X(\infty), we investigate the distribution of the set {(y\gamma,b\gamma^{-1}):\gamma\in\Gamma} in X\times X(\infty). It is proved, in particular, that the orbits of \Gamma in the Furstenberg boundary are equidistributed, and that the orbits of \Gamma in X are equidistributed in ``sectors'' defined with respect to a Cartan decomposition. We also discuss an application to the Patterson-Sullivan theory. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces. 
Alexander Gorodnik and Hee Oh