Research of automorphisms preserving geometric structure of manifolds

点击翻译按钮获取中文摘要

In this reseach we study the group of diffeomorphisms and Lipschitz homeomorphisms of smooth manifolds and equivariant diffeomorphisms of smooth G-manifolds. We also study pseudo Anosov homeomorphisms on the surfaces. We have the following results.(1) Let D(M) denote the group of diffeomorphisms of a smooth manifold M which are isotopic to the identity through diffeomorphisms with compact support. Then it is known that D(M) is perfect. We proved that D(M) is perfect as well when M is a manifold with boundary of dimension greater than one. We applied the result to prove that D_G(M) is perfect when M is the Hirzebruch-Mayer O(n)-manifold. Here D_G(M) denote the group of equivariant diffeomorphisms of M which are G-isotopic to the identity through diffeomorphisms with compact support.(2) Let V be a representation space of a finite group G. Using the linealization theorem by Sternberg and perfectness theorem by Tsuboi, we calculated the first homology group H_1(D_G(M)). We can apply the result to calculate H_1(D(M)) when M is a smooth orbifold.(3) LetΓ=SL(2,Z) denote the modular group which acts on the half plane H canonically. We showed H_1(D(H/Γ)) is related to elliptic fixed point set of Γ. Let H* denote the set H adding the cusp points of Γ. We proved that H_1(D(H/Γ)) is related to the elliptic fixed point set of Γ and also the cusp point set of Γ.(4) There is the problem to determine the minimal value of the dilatation of pseudo Anosov homeomorphisms of the oriented surface of genus g. We found important examples to estimate the minimal value of the dilatations for each genus g with respect to the known examples. The method is investigating the Birkov cross section of the Anosov flow.(5) We held the conference on diffeomorphism and the related fields partially supported by Grant-in-Aid for Scientific Research (http://math.shinshu-u.ac.jp/~kabe/diffeo-program.htm).