The necessary and sufficient condition for the group of leaf preserving diffeomorphisms to be simple

Abstract

Let F be a C^∞-foliation on a compact C^∞-manifold M. We consider the group of all leaf preserving C^∞-diffeomorphisms of (M, F) which are isotopic to the identity through leaf preserving C^∞-diffeomorphisms. Then we show that the group is simple if and only if all leaves of F are dense.