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

Abstract

Let X be a symmetric space of noncompact type, and let Γ be a lattice in the isometry group of X. We study the distribution of orbits of Γ acting on the symmetric space X and its geometric boundary X(∞), generalizing the main equidistribution result of Margulis’s thesis [M, Theorem 6] to higher-rank symmetric spaces. More precisely, for any y ∈ X and b ∈ X(∞), we investigate the distribution of the set {(yγ, bγ^(−1)) : γ ∈ } in X × X(∞). It is proved, in particular, that the orbits of Γ in the Furstenberg boundary are equidistributed and that the orbits of Γ in X are equidistributed in “sectors” defined with respect to a Cartan decomposition. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces, which we obtain using Shah’s result [S, Corollary 1.2] based on Ratner’s measure-classification theorem [R1, Theorem 1].