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Asymptotic properties of linear groups. II. (Propriétés asymptotiques des groupes linéaires. II.) (French) Zbl 0960.22012
Kobayashi, Toshiyuki (ed.) et al., Analysis on homogeneous spaces and representation theory of Lie groups. Based on activities of the RIMS Project Research ’97, Okayama-Kyoto, Japan, during July and August 1997. Tokyo: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 26, 33-48 (2000).
[Part I by the same author in Geom. Funct. Anal. 7, 1-47 (1997).]
Let $$G$$ be a connected real semisimple linear group with Lie algebra $${\mathcal G}$$. Let $$G=K\exp ({\mathcal A}^+)K$$ be a Cartan decomposition of $$G$$. Denote by $$m:G\to {\mathcal A}^+$$ the map associated with this decomposition and let $$\Gamma$$ be a Zariski dense subgroup of $$G$$. Let $$l_\Gamma$$ be the cone asymptotic to $$m(\Gamma)$$. For every $$\eta>0$$ and every closed cone $$C$$ such that $$C-\{0\}$$ is included in the interior of $$l_\Gamma$$, every point of $$C$$ outside a compact is at distance less than $$\eta$$ from $$m(\Gamma)$$.
For the entire collection see [Zbl 0941.00016].

##### MSC:
 2.2e+47 Semisimple Lie groups and their representations 2.2e+16 General properties and structure of real Lie groups