On the group of real analytic diffeomorphisms

Tsuboi, Takashi

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 4, pp. 601-651

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the $n$-dimensional torus, its identity component is a simple group. For $U\left(1\right)$ fibered manifolds, for manifolds admitting special semi-free $U\left(1\right)$ actions and for 2- or 3-dimensional manifolds with nontrivial $U\left(1\right)$ actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

Le groupe des difféomorphismes analytiques réels d’une variété analytique réelle est un groupe riche. Il est dense dans le groupe des difféomorphismes lisses. Herman a montré que, pour le tore de dimension $n$, sa composante connexe de l’identité est un groupe simple. Pour les variétés $U\left(1\right)$ fibrées, pour les variétés admettant une action semi-libre spéciale de $U\left(1\right)$, et pour les variétés de dimension $2$ ou $3$ admettant une action non-triviale de $U\left(1\right)$, on montre que la composante de l’identité du groupe des difféomorphismes analytiques réels est un groupe parfait.

Classification : 57R52, 57R50, 58A07, 58F18, 57R30, 53C12, 58C15, 37C05
Keywords: diffeomorphism groups, foliations, real analytic, rotations, $U(1)$ action, circle bundles
Mots-clés : groupes de difféomorphismes, feuilletages, analytique réel, rotations, action de $U(1)$, fibrés en cercle

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