Françoise Dal'bo

Séries de poincaré des groupes géométriquement finis

In this paper, we study the behaviour of the Poincaré series of a geometrically finite group Γ of isometries of a riemannian manifoldX with pinched curvature, in the case when Γ contains parabolic elements. We give a sufficient condition on the parabolic subgroups of Γ in order that Γ be of divergent type. When Γ is of divergent type, we show that the Sullivan measure on the unit tangent bundle ofX/Γ is finite if and only if certain series which involve only parabolic elements of Γ are convergent. We build also examples of manifoldsX on which geometrically finite groups of convergent type act.

A criterion of conjugacy for Zariski dense subgroups

Abstract Let G 1 , G 2 be simple Lie groups with trivial center and Γ 1 , Γ 2 Zariski dense subgroups of G 1 , G 2 . We show that, if ϕ is a surjective homomorphism between Γ 1 and Γ 2 , then ϕ can be extended to an isomorphism between G 1 and G 2 if and only if the group {( γ , ϕ ( γ ))∣ γ ∈ Γ 1 } is not Zariski dense in G 1 × G 2 . Using this criterion we prove the marked length rigidity of Zariski dense groups acting on symmetric spaces G / K , where G is a noncompact simple Lie group with trivial center and K is a maximal compact subgroup of G . This result also holds for semisimple groups of noncompact type.

On the growth of nonuniform lattices in pinched negatively curved manifolds

We study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices. These objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold .