Propriétés Asymptotiques des Groupes Linéaires (II)

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 $m(\Gamma)$.