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

摘要

Let X be a symmetric space of noncompact type, and let Gamma be 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(infinity), generalizing the main equidistribution result of Margulis's thesis [M, Theorem 6] to higher-rank symmetric spaces. More precisely, for any y is an element of X and b is an element of X(infinity), we investigate the distribution of the set 1(y gamma, b gamma(-1)) : y c F) in X x X(infinity). 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. 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].