## Duke Mathematical Journal

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

#### Abstract

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

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

Dates
First available in Project Euclid: 24 August 2007

http://projecteuclid.org/euclid.dmj/1187916268

Digital Object Identifier
doi:10.1215/S0012-7094-07-13933-4

Mathematical Reviews number (MathSciNet)
MR2350851

Zentralblatt MATH identifier
1132.22012

Subjects