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15 September 2007 Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary
Alexander Gorodnik, Hee Oh
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Duke Math. J. 139(3): 483-525 (15 September 2007). DOI: 10.1215/S0012-7094-07-13933-4

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 yX and bX(), we investigate the distribution of the set {(yγ,bγ1):γΓ} 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]

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Alexander Gorodnik. Hee Oh. "Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary." Duke Math. J. 139 (3) 483 - 525, 15 September 2007. https://doi.org/10.1215/S0012-7094-07-13933-4

Information

Published: 15 September 2007
First available in Project Euclid: 24 August 2007

zbMATH: 1132.22012
MathSciNet: MR2350851
Digital Object Identifier: 10.1215/S0012-7094-07-13933-4

Subjects:
Primary: 22E40
Secondary: 37A17

Rights: Copyright © 2007 Duke University Press

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Vol.139 • No. 3 • 15 September 2007
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