Duke Mathematical Journal

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

Alexander Gorodnik and Hee Oh

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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 $X(\infty)$, generalizing the main equidistribution result of Margulis's thesis [M, Theorem 6] to higher-rank symmetric spaces. More precisely, for any $y\in X$ and $b\in X(\infty)$, we investigate the distribution of the set $\{(y\gamma, b\gamma^{-1}):\gamma \in \Gamma\}$ in $X\times X(\infty)$. 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]

Article information

Duke Math. J. Volume 139, Number 3 (2007), 483-525.

First available in Project Euclid: 24 August 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 37A17: Homogeneous flows [See also 22Fxx]


Gorodnik, Alexander; Oh, Hee. Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary. Duke Math. J. 139 (2007), no. 3, 483--525. doi:10.1215/S0012-7094-07-13933-4. http://projecteuclid.org/euclid.dmj/1187916268.

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