Simplicité des groupes de transformations de surfaces

SOCIEDADE BRASILEIRA DE MATEMÁTICA ENSAIOS MATEM´
ATICOS
2008, Volume 14, 1–143
Simplicit´e des groupes de
transformations de surfaces
Abed Bounemoura
Abstract. The study of algebraic properties of groups of transformations
of a manifold gives rise to an interplay between different 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 complete 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 folia-
tions, and a result of Mather showing that the theorem remains true for
Ck-diffeomorphisms, provided that kis different from n+ 1, where nis
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 conservative diffeomorphisms, the situation is well unders-
tood thanks to the work of Banyaga, but this is definitely not the case for
conservative homeomorphisms of surfaces, and we will present some open
problems in this direction as well as different attempts to solve them.
2000 Mathematics Subject Classification: 58D05, 37E30.

Remerciements
Ce texte est issu d’un stage effectu´e sous la direction de Fr´ed´eric Le
Roux `a l’universit´e d’Orsay. Je le remercie tr`es profond´ement de m’avoir
fait confiance en acceptant de me guider sur ce travail, ainsi que de la gen-
tillesse et de la disponibilit´e dont il a fait preuve. Sans ses explications,
ses nombreuses am´eliorations et ses multiples relectures approfondies, la
r´edaction de ce manuscrit n’aurait tout simplement pas ´et´e possible. Il m’a
´egalement beaucoup aid´e `a la fin pour terminer la r´edaction de l’introduc-
tion alors que j’´etais en panne. Je me permets encore de lui exprimer toute
ma gratitude.
Je remercie ´egalement ´ Etienne Ghys pour l’int´erˆet qu’il a port´e `a ce
travail, Fran¸cois B´eguin, Sylvain Crovisier, Pierre Py et J´erˆome Buzzi pour
leurs relectures et leurs commentaires sur certaines parties de ce texte, ainsi
que tous mes amis doctorants d’Orsay et de Chevaleret.

Table des mati`eres
Introduction 7
1 Hom´eomorphismes de surfaces 11
1.1 Simplicit´e du groupe Homeo0(M) .............. 11
1.2 Simplicit´e du groupe Homeo0(M) .............. 15
2 Diff´eomorphismes de surfaces 22
2.1 Simplicit´e du groupe Diff∞
0(T2)............... 22
2.2 Simplicit´e du groupe Diff∞
0(M)............... 30
2.3 Feuilletages et espaces classifiants . . . . . . . . . . . . . . . 43
2.4 Simplicit´e de Diff r
0(M) pour r6=3 ............. 53
3 Diff´eomorphismes conservatifs 62
3.1 Premier invariant de Calabi . . . . . . . . . . . . . . . . . . 63
3.2 Diff´eomorphismes hamiltoniens . . . . . . . . . . . . . . . . 72
3.3 Second invariant de Calabi . . . . . . . . . . . . . . . . . . . 81
3.4 Th´eor`eme de simplicit´e de Banyaga . . . . . . . . . . . . . . 84
4 Hom´eomorphismes conservatifs 98
4.1 Approche de Fathi . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Approche de Gambaudo et Ghys . . . . . . . . . . . . . . . 105
4.3 Distance de Hofer . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4 ApprochedeOh ........................ 119
A Th´eor`eme de Schoenflies 125
B Th´eor`eme d’Epstein 128
C Th´eor`eme de Nash-Moser-Hamilton 133
Bibliographie 139