Commutator Length of Leaf Preserving Diffeomorphisms

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 $\mathcal{F}$ on the torus is uniformly perfect if and only if $\mathcal{F}$ 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 $n\ge 2$.