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.