G. Courtois - The Margulis lemma, old and new (Part 3)

The Margulis lemma describes the structure of the group generated by small loops in the fundamental group of a Riemannian manifold, thus giving a picture of its local topology. Originally stated for homogeneous spaces by C. Jordan, L. Bieberbach, H. J. Zassenhaus, D. Kazhdan-G. Margulis, it has been extended to the Riemannian setting by G. Margulis for manifolds of non positive curvature. The goal of these lectures is to present the recent work of V. Kapovitch and B. Wilking who gave a sharp version of the Margulis lemma under the assumption that the Ricci curvature is bounded below. Their method uses the structure of « Ricci limit spaces » explained by T. Richard during his lectures.