Geometric study on infinite simple groups

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We studied infinite simple groups as geometric objects where they have verious quasimorphisms and the distance function d on the set of unions of a conjugate class and its inverse. We determined the quasi isometry type with respect to the distance d of the infinite alternative group A_infty. We showed and published the result that the identity component of the group of real analytic diffeomorphisms of a manifold with good circle actions is perfect. We showed and published the result that the identity component of the group of diffeomorphisms of a closed connected manifold of dimensions not eaual to 2 and 4 is uniformly perfect and uniformly simple. We also studied the homeomorphism groups of commutator width one.