On the Perfectness of Nontransitive Groups of Diffeomorphisms


. It is proven that the identity component of the group preserving the leaves of a generalized foliation is perfect. This shows that a well-known simplicity theorem on the diffeomorphism group extends to the nontransitive case. 1. Introduction The goal of this paper is to show a perfectness theorem for a class of diffeomorphism groups connected with foliations. Throughout by foliations we shall understand generalized foliations in the sense of P.Stefan [12]. Given a smooth foliation F on a manifold M , the symbol Diff(T n ; F) 0 stands for the identity component of the group of all leaf preserving C 1 -smooth diffeomorphisms of (M;F) with compact support. Theorem 1.1. Let F be a foliation on M with no leaves of dimension 0. Then the group Diff(T n ; F) 0 and its universal covering Diff(T n ; F) 0 are perfect. Theorem 1.1 and its proof extend a well known theorem of W.P.Thurston [15] stating that Diff(M) 0 , the identity component of the group of all compactly supported dif..

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