JournalscmhVol. 87, No. 1pp. 141–185

On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds

  • Takashi Tsuboi

    University of Tokyo, Japan
On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds cover
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Abstract

We show that the identity component Diffr(M2m)0^r(M^{2m})_0 of the group of CrC^r diffeomorphisms of a compact (2m)(2m)-dimensional manifold M2mM^{2m} (1r1\leq r\leq \infty, r2m+1r\neq 2m+1) is uniformly perfect for 2m62m\geq 6, i.e., any element of Diffr(M2m)0^r(M^{2m})_0 can be written as a product of a bounded number of commutators. It is also shown that for a compact connected manifold M2mM^{2m} (2m62m\geq 6), the identity component Diffr(M2m)0^r(M^{2m})_0 of the group of CrC^r diffeomorphisms of M2mM^{2m} (1r1\leq r\leq \infty, r2m+1r\neq 2m+1) is uniformly simple, i.e., for elements ff and gg of Diffr(M2m)0{^r(M^{2m})_0\setminus \{id}\}, ff can be written as a product of a bounded number of conjugates of gg or g1g^{-1}.

Cite this article

Takashi Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds. Comment. Math. Helv. 87 (2012), no. 1, pp. 141–185

DOI 10.4171/CMH/251