# Dynamique topologique d'une action de groupe sur un espace homogène : exemples d'actions unipotente et diagonale

Abstract : This thesis deals with two examples of group actions on homogeneous spaces and their topological dynamics. Each of them is conjugate to a flow on a fibered space over a locally symmetric space. The first chapter contains generalities about hyperbolic spaces, their products and their isometries groups. The second chapter is devoted to the action of the upper unipotent subgroup on the quotient of the projective unimodular complex $2\times2$ group by a discrete subgroup. This action is conjugate to a frame flow on the tangent bundle of an hyperbolic 3-manifold. Dense and closed orbits are characterised. Hence we obtain a dynamical characterisation of different classes of Kleinian groups (geometrically finite, convex-cocompact and lattices). In the third chapter, we consider the product of two copies of the real projective unimodular $2\times2$ group and study the action of the product of diagonal subgroups on finite-volume quotients. When such a quotient is irreducible, Margulis conjectured that every orbit is either dense or closed. Closed orbits are characterised and we exhibit points of the Furstenberg boundary giving rise to dense orbits. In the last chapter, we look more closely at the special case of Hilbert modular lattices. We study the relation between the above conjecture about orbits and the diophantine approximation of pairs of real numbers by a real quadratic field.
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Theses
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https://tel.archives-ouvertes.fr/tel-00007213
Contributor : Marie-Annick Guillemer <>
Submitted on : Wednesday, October 27, 2004 - 3:34:43 PM
Last modification on : Friday, July 10, 2020 - 4:04:15 PM
Long-term archiving on: : Friday, April 2, 2010 - 8:27:42 PM

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• HAL Id : tel-00007213, version 1

### Citation

Damien Ferte. Dynamique topologique d'une action de groupe sur un espace homogène : exemples d'actions unipotente et diagonale. Mathématiques [math]. Université Rennes 1, 2003. Français. ⟨tel-00007213⟩

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