Correspondence between diffeomorphism groups and singular foliations

2011/03/18 by Tomasz Rybicki, Rybicki, Tomasz
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1103.3623

9 pages

arxiv created 2011/03/18 · arxiv updated 2011/03/21

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^∞-diffeomorphism group of a regular foliation \F is simple iff \F has no proper minimal sets.

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