# Commutator Length of Leaf Preserving Diffeomorphisms

### Kazuhiko Fukui

Kyoto Sangyo University, Japan

## Abstract

We consider the group of leaf preserving $C^{\infty}$-diffeomorphisms for a $C^{\infty}$-foliation on a manifold which is isotopic to the identity through leaf preserving $C^{\infty}$-diffeomorphisms with compact support. Then we show that the group for a one-dimensional $C^{\infty}$-foliation ${\cal F}$ on the torus is uniformly perfect if and only if ${\cal F}$ has no compact leaves. Moreover we consider the group of leaf preserving $C^{\infty}$-diffeomorphisms for the product foliation on $S^1 \times S^n$ which is isotopic to the identity through leaf preserving $C^{\infty}$-diffeomorphisms. Here the product foliation has leaves of the form $\{ pt \} \times S^n$. Then we show that the group is uniformly perfect for $n \ge 2$.

## Cite this article

Kazuhiko Fukui, Commutator Length of Leaf Preserving Diffeomorphisms. Publ. Res. Inst. Math. Sci. 48 (2012), no. 3, pp. 615–622

DOI 10.2977/PRIMS/83