Study on Diffeomorphism Groups preserving a Geometric Structure

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In order to clarify geometric structures on a manifold, I studied the algebraic structure of the diffeomorphism group preserving a geometric structure. As results, I got (1) characterization of the simplicity of the leaf preserving diffeomorphism group for foliated manifolds, (2) consideration of commutator length for leaf preserving diffeomorphisms, especially characterization of the uniform perfectness of the groups for 1 dimensional foliations on the 2-torus and (3) for manifold pair (M,N), characterization of the uniform perfectness of the diffeomorphism group D(M,N) preserving N.