Simplicité des groupes de transformations de surfaces

Abstract. The study of algebraic properties of groups of transformations of a manifold gives rise to an interplay between différent areas of mathe- matics such as topology, geometry, dynamical Systems and foliation theory. This volume is devoted to the question of simplicity of such groups, and we will mainly restrict our attention to the case where the manifold is a surface. In the first chapter, we will show that the identity component of the group of homeomorphisms of a closed surface is simple. This will lead us to the case of diffeomorphisms, and in the second chapter, we will give the complète proof of the Epstein-Herman-Mather-Thurston theorem stating that the group of C°° -diffeomorphisms isotopic to the identity is also simple. We will also review the link with classifying spaces for foliations, and a resuit of Mather showing that the theorem remains true for Cfe-diffeomorphisms, provided [...]that k is différent from n + 1, where n is the dimension of the manifold. The last two chapters deal with conserva- tive homeomorphisms and diffeomorphisms, by which we mean preserving a measure or a smooth volume or symplectic form. In those cases, there is a generalized rotation number showing that the associated groups cannot be simple. For conservâtive diffeomorphisms, the situation is well unders- tood thanks to the work of Banyaga, but this is dennitely not the case for conservât ive homeomorphisms of surfaces, and we will présent some open problems in this direction as well as différent attempts to solve them.