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Correspondence between diffeomorphism groups and singular foliations

Tom 103 / 2012

Tomasz Rybicki Annales Polonici Mathematici 103 (2012), 27-35 MSC: Primary 57R50; Secondary 53C12, 57R52. DOI: 10.4064/ap103-1-3

Streszczenie

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.

Autorzy

  • Tomasz RybickiFaculty of Applied Mathematics
    AGH University of Science and Technology
    Al. Mickiewicza 30
    30-059 Kraków, Poland
    e-mail

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