Correspondence between diffeomorphism groups and singular foliations
Volume 103 / 2012
Abstract
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 $C^{\infty}$ diffeomorphism group of a regular foliation $\mathcal F$ is simple iff $\mathcal F$ has no proper minimal sets.
