## Advanced Studies in Pure Mathematics

### Propriétés Asymptotiques des Groupes Linéaires (II)

Yves Benoist

#### Abstract

Let $G = K exp({\mathfrak{a}}^+)K$ be a Cartan decomposition of a connected real linear semisimple Lie group and $m : G \to {\mathfrak{a}}^+$ be the associated map. Let $\Gamma$ be a Zariski dense subgroup of $G$ and $\ell_\Gamma$ be the asymptotic cone to $m(\Gamma)$. This cone is convex and of non empty interior (cf [3]).

We show that $m(\Gamma)$ fills completely $\ell_\Gamma$ in the following sense: for every $\varepsilon \gt 0$ and every closed cone $C$ such that $C - \{0\}$ is included in the interior of $\ell_\Gamma$, every point of $C$ outside a compact is at distance less than $\varepsilon$ from $m(\Gamma)$.

#### Résumé

Soient $G = K exp({\mathfrak{a}}^+)K$ une décomposition de Cartan d'un groupe linéaire semisimple réel connexe, $m : G \to {\mathfrak{a}}^+$ l'application correspondante, $\Gamma$ un sous-semigroupe Zariski dense de $G$ et $\ell_\Gamma$ le cône asymptote à $m(\Gamma)$. Ce cône est convexe et d'intérieur non vide (cf [3]).

Nous montrons en quel sens $m(\Gamma)$ remplit complètement ce cône $\ell_\Gamma$.

#### Article information

Dates
Received: 1 December 1997
First available in Project Euclid: 20 August 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1534788122

Digital Object Identifier
doi:10.2969/aspm/02610033

Mathematical Reviews number (MathSciNet)
MR1770716

Zentralblatt MATH identifier
0960.22012

#### Citation

Benoist, Yves. Propriétés Asymptotiques des Groupes Linéaires (II). Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto, 33--48, Mathematical Society of Japan, Tokyo, Japan, 2000. doi:10.2969/aspm/02610033. https://projecteuclid.org/euclid.aspm/1534788122