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

### Takashi Tsuboi

University of Tokyo, Japan

## 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 $M_{2m}$ ($1≤r≤∞$, $r=2m+1$) is uniformly perfect for $2m≥6$, 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 $M_{2m}$ ($2m≥6$), the identity component Diff$_{r}(M_{2m})_{0}$ of the group of $C_{r}$ diffeomorphisms of $M_{2m}$ ($1≤r≤∞$, $r=2m+1$) is uniformly simple, i.e., for elements $f$ and $g$ of Diff$_{r}(M_{2m})_{0}∖{$id$}$, $f$ can be written as a product of a bounded number of conjugates of $g$ or $g_{−1}$.

## 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