Abstract. The notion of a C r,s -diffeomorphism related to a foliation is introduced. A perfectness theorem for the group of C r,s -diffeomorphisms is proved. A remark on C n+1 -diffeomorphisms is given.

Abstract. Given a principal G-bundle π : M → B, let HG(M ) be the identity component of the group of G-equivariant homeomorphisms on M . The problem of the uniform perfectness and boundedness of HG(M ) is studied. It occurs that these properties depend on the structure of H(B), the identity component of the group of homeomorphisms of B, and of B itself. Most of the obtained results still hold in the C r category.

Let M be a manifold, p ∈ M and let H(M, p) be the identity component of the group of all compactly supported homeomorphisms of M fixing p. It is shown that H(M, p) is a perfect group. Next, we prove that the group H(R n , 0) is bounded. In contrast, in the C ∞ category the diffeomorphism group D ∞ (R n , 0), analogous to H(R n , 0), is neither perfect nor bounded. Finally, the boundedness and uniform perfectness of H(M, p) is studied.

Abstract. It is shown that in some generic cases the identity component of the group of leaf preserving diffeomorphisms (with not necessarily compact support) on a foliated open manifold is perfect. Next, it is proved that it is also bounded, i.e. bounded with respect to any bi-invariant metric. It follows that the group is uniformly perfect as well.

Abstract. It is shown that the identity component of the group of all compactly supported C°°-diffeomorphisms of a manifold with corners is perfect provided the manifold has no vertices. An analogous result for C r -diffeomorphisms still holds whenever r > ra+1, where n is the dimension of the manifold. These results constitute a generalization of a well-known theorem by Thurston and Mather on the simplicity of diffeomorphism groups of boundary less manifolds.