I will present a recent result in the theory of unitary representations of lattices in semi-simple Lie groups, which can be viewed as simultaneous generalization of Margulis normal subgroup theorem and C*-simplicity and the unique trace property for such lattices. The strategy of proof gathers ideas of both of these results: we extend Margulis' dynamical approach to the non-commutative setting, and apply this to the conjugation dynamical system induced by a unitary representation. On the way, we obtain a new proof of Peterson's character rigidity result, and a new rigidity result for uniformly recurrent subgroups of such lattices. I will give some basics on non-commutative ergodic theory and explain-some steps to prove the main result and its applications. This is based on joint works with Uri Bader, Cyril Houdayer, and Jesse Peterson.[-]

Viewed as the boundary at infinity of the complex hyperbolic plane, the 3-sphere is equipped with a contact structure. The interplay between this contact structure and limit sets of subgroups of PU(2,1) has deep consequences on the properties of these subgroups. Some limit sets enjoy the property of slimness, that we introduce. Using this property, one can shed new lights on known results, and better describe deformations of subgroups. I will present this notion through simple examples and pictures, and then describe some results we obtain. This is a joint work with E. Falbel and P. Will.[-]