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On the uniform perfectness of equivariant diffeomorphism groups for principal g manifolds

Fukui K.

Opuscula Mathematica

2017

Vol. 37, no. 3

381--388

We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978), 52–54] that the identity component [formula] of the group of equivariant Cr-diffeomorphisms of a principal G bundle M over a manifold B is perfect for a compact connected Lie group G and [formula] In this paper, we study the uniform perfectness of the group of equivariant Cr-diffeomorphisms for a principal G bundle M over a manifold B by relating it to the uniform perfectness of the group of Cr-diffeomorphisms of B and show that under a certain condition, [formula] is uniformly perfect if B belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant Cr-diffeomorphisms for principal G bundles over closed manifolds of dimension less than or equal to 3, and in particular we prove the uniform perfectness of the group for the 3-dimensional case and r ≠ 4.

On the structure of certain nontransitive diffeomorphism groups on open manifolds

Kowalik A., Lech J., Michalik I.

Opuscula Mathematica

2012

Vol. 32, no. 3

511-520

It is shown that in some generic cases the identity component of the group of leaf preserving diffeomorphisms (with not necessarily compact support) on a foliated open manifold is perfect. Next, it is proved that it is also bounded, i.e. bounded with respect to any bi-invariant metric. It follows that the group is uniformly perfect as well.