Correspondence between diffeomorphism groups and singular foliations

Tomasz Rybicki¹

¹ Faculty of Applied Mathematics AGH University of Science and Technology Al. Mickiewicza 30 30-059 Kraków, Poland

Annales Polonici Mathematici, Tome 103 (2012) no. 1, pp. 27-35.

Résumé

It is well-known that any isotopically connected diffeomorphism group $G$ of a manifold determines a unique singular foliation ${\mathcal{F}}_G$. A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup $[G,G]$ of an isotopically connected, factorizable and non-fixing $C^r$ diffeomorphism group $G$ is simple iff the foliation ${\mathcal{F}}_{[G,G]}$ defined by $[G,G]$ admits no proper minimal sets. In particular, the compactly supported $e$-component of the leaf preserving $C^{\mathrm{\infty }}$ diffeomorphism group of a regular foliation $\mathcal{F}$ is simple iff $\mathcal{F}$ has no proper minimal sets.

Détail

DOI : 10.4064/ap103-1-3

Mots-clés : well known isotopically connected diffeomorphism group manifold determines unique singular foliation mathcal one to one correspondence between class singular foliations subclass diffeomorphism groups established illustration correspondence shown commutator subgroup isotopically connected factorizable non fixing diffeomorphism group simple foliation mathcal defined admits proper minimal sets particular compactly supported e component leaf preserving infty diffeomorphism group regular foliation mathcal simple mathcal has proper minimal sets