Correspondence between diffeomorphism groups and singular foliations

TL;DR

This paper establishes a one-to-one correspondence between singular foliations and a subclass of diffeomorphism groups, revealing conditions under which certain groups are simple based on properties of the associated foliations.

Contribution

It introduces a bijective relationship between singular foliations and specific diffeomorphism groups, and characterizes the simplicity of these groups via minimal sets of the foliations.

Findings

The commutator subgroup of certain diffeomorphism groups is simple iff the associated foliation has no proper minimal sets.

The leaf-preserving diffeomorphism group component is simple iff the foliation admits no proper minimal sets.

A one-to-one correspondence between singular foliations and a subclass of diffeomorphism groups is established.

Abstract

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