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

Abstract

We show that the identity component Diff$^r(M^{2m})_0$ of the group of $C^r$ diffeomorphisms of a compact $(2m)$-dimensional manifold $M^{2m}$ ($1\leq r\leq \infty$, $r\neq 2m+1$) is uniformly perfect for $2m\geq 6$, i.e., any element of Diff$^r(M^{2m})_0$ can be written as a product of a bounded number of commutators. It is also shown that for a compact connected manifold $M^{2m}$ ($2m\geq 6$), the identity component Diff$^r(M^{2m})_0$ of the group of $C^r$ diffeomorphisms of $M^{2m}$ ($1\leq r\leq \infty$, $r\neq 2m+1$) is uniformly simple, i.e., for elements $f$ and $g$ of Diff$^r(M^{2m})_0\setminus \{$id$\}$, $f$ can be written as a product of a bounded number of conjugates of $g$ or $g^{-1}$.