Study on Diffeomorphism Groups and Commutator Length

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I studied the algebraic structure (for instance, 1 dimensional homology group and commutator length and others) of the diffeomorphism groups preserving geometric structures. As results, I got (1) Decision of 1 dimensional homology group of the equivariant diffeomorphism groups for representation spaces of finite groups and its application to various geometric structures, (2) Decision of 1 dimensional homology group of the foliation preserving diffeomorphism groups for foliations with singularities of Morse type, (3) Proof of the perfectness of the diffeomorphism group preserving a submanifold and the consideration of the uniform perfectness for special compact manifolds with boundary, (4) Characterization of the simplicity of the leaf preserving diffeomorphism group for foliated manifolds, and (5) Characterization of the uniform perfectness of the foliation preserving diffeomorphism groups for 1 dimensional foliations on the torus.