JournalsprimsVol. 48, No. 3pp. 615–622

Commutator Length of Leaf Preserving Diffeomorphisms

  • Kazuhiko Fukui

    Kyoto Sangyo University, Japan
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Abstract

We consider the group of leaf preserving CC^{\infty}-diffeomorphisms for a CC^{\infty}-foliation on a manifold which is isotopic to the identity through leaf preserving CC^{\infty}-diffeomorphisms with compact support. Then we show that the group for a one-dimensional CC^{\infty}-foliation F{\cal F} on the torus is uniformly perfect if and only if F{\cal F} has no compact leaves. Moreover we consider the group of leaf preserving CC^{\infty}-diffeomorphisms for the product foliation on S1×SnS^1 \times S^n which is isotopic to the identity through leaf preserving CC^{\infty}-diffeomorphisms. Here the product foliation has leaves of the form {pt}×Sn\{ pt \} \times S^n. Then we show that the group is uniformly perfect for n2n \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