## Duke Mathematical Journal

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

### 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

**Permanent link to this document**

https://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**

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

Secondary: 37A17: Homogeneous flows [See also 22Fxx]

#### Citation

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