Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary

Abstract

Let X be a symmetric space of noncompact type and Γa lattice in the isometry group of X. We study the distribution of orbits of Γacting 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 {(yγ,bγ^{-1}):γ\inΓ} in X\times X(\infty). It is proved, in particular, that the orbits of Γin the Furstenberg boundary are equidistributed, and that the orbits of Γin 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.