Correspondence between diffeomorphism groups and singular foliations

It is well-known that any isotopically connected diffeomorphism group $G$ of a manifold determines uniquely a singular foliation ${\text{F}}_G$. 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 ${\text{F}}_{[G,G]}$ 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 $\text{F}$ is simple iff $\text{F}$ has no proper minimal sets.