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Published September 15, 2007 | Published
Journal Article Open

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].

Additional Information

© 2007 Duke University Press. Received 6 September 2006. Revision received 12 December 2006. Gorodnik's work partially supported by National Science Foundation grant DMS-0400631. Oh's work partially supported by National Science Foundation grant DMS-0333397. Gorodnik thanks the Department ofMathematics at the California Institute of Technology for its hospitality during August 2003, when most of this work was done. We thank François Ledrappier for helpful remarks on the preliminary version of our article.

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