Commutator length of leaf preserving diffeomorphisms (Q713088)

Summary: 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 \(\{ \text{pt} \} \times S^n\). Then we show that the group is uniformly perfect for \(n \geq 2\).