Propriétés Asymptotiques des Groupes Linéaires

Abstract.

Let G be a reductive linear real Lie group and $\Gamma$ be a Zariski dense subgroup. We study asymptotic properties of $\Gamma$ through the set of logarithms of the radial components of the elements of $\Gamma$: we prove that the asymptotic cone of this set is a convex cone with non empty interior and is stable by the Cartan involution. Reciprocally any closed convex cone of the positive Weyl chamber whose interior is non empty and which is stable by the opposition involution can be obtained this way.¶We relate this limit cone and the limit set of $\Gamma$ to the set of open semigroups of G which meet $\Gamma$.¶We also prove similar results over any local fields.