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Research of automorphisms preserving geometric structure of manifolds

Research Project

Project/Area Number 16540058
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionShinshu University

Principal Investigator

ABE Kojun  Shinshu University, Faculty of Science, Professor, 理学部, 教授 (30021231)

Co-Investigator(Kenkyū-buntansha) MINAKAWA Hiroyuki  Yamagata University, Faculty of Education, Associate Professor, 教育学部, 助教授 (30241300)
Project Period (FY) 2004 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Keywordsdiffeomorphism group / Lipschitz homeomorphism group / pseudo Anosov homeomorphism / smooth orbifold / Teichmuller space / first homology / equivariant diffeomorphism group / smooth vector field / Lipschitz同相写像 / 可微分G-多様体 / Lischitz同相写像
Research Abstract

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).

Report

(3 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • Research Products

    (13 results)

All 2006 2005 2004

All Journal Article (13 results)

  • [Journal Article] On the first homology of the group of equivariant Lipschitz homeomorphisms2006

    • Author(s)
      Kojun Abe, Kazuhiko Fukui, Takeshi Miura
    • Journal Title

      Jour. Math. Soc. Japan 58・1

      Pages: 1-15

    • NAID

      10017178257

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the first homology of the group of equivariant Lipschitz homeomorphisms2006

    • Author(s)
      Kojun Abe, Kazuhiko Fukui, Takeshi Miura
    • Journal Title

      Jour.Math.Soc.Japan 58・1

      Pages: 1-15

    • NAID

      10017178257

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] On the first homology of automorphism groups of manifolds with geometric structures2005

    • Author(s)
      Kojun Abe, Kazuhiko Fukui, Takeshi Miura
    • Journal Title

      Central European Jour. Math. 3

      Pages: 516-528

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the diffeomorphism group of a smooth orbifold and it's applications2005

    • Author(s)
      阿部 孝順
    • Journal Title

      数理解析研究所講究録 1449

      Pages: 1-11

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the first homology of automorphism groups of manifolds with geometric structures2005

    • Author(s)
      Kojun Abe, Kazuhiko Fukui
    • Journal Title

      Central European Jour.Math. 3

      Pages: 516-528

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the diffeomorphism group of a smooth orbifold and its applications2005

    • Author(s)
      Kojun Abe
    • Journal Title

      Surikaisekikenkyusho Kokyuroku 1449

      Pages: 1-11

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the first homology of automorphism groups of manifolds with g eometric structures2005

    • Author(s)
      Kojun Abe, Kazuhiko Fukui
    • Journal Title

      Central European Jour.Math. 3

      Pages: 516-528

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the diffeomorphism group of a smooth orbifold and it' s applications2005

    • Author(s)
      阿部 孝順
    • Journal Title

      数理解析研究所講究録 1449

      Pages: 1-11

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Algebraic property of dilatation constants of piecewise linear structures of Anosov foliations2004

    • Author(s)
      Hiroyuki Minakawa
    • Journal Title

      Jour. Math. Sci. Univ. Tokyo 11

      Pages: 65-74

    • NAID

      110000071245

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 可微分軌道体上のベクトル場の構造と多項式写像の特異点2004

    • Author(s)
      阿部 孝順
    • Journal Title

      数理解析研究所講究録 1374

      Pages: 95-108

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Algebraic property of dilatation constants of piecewise linear structures of Anosov foliations2004

    • Author(s)
      Hiroyuki Minakawa
    • Journal Title

      Jour.Math.Sci.Univ.Tokyo 11

      Pages: 65-74

    • NAID

      110000071245

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the structure of smooth vector fields on smooth orbifolds and singularities of polynomial maps2004

    • Author(s)
      Kojun Abe
    • Journal Title

      Surikaisekikenkyusho Kokyuroku 1374

      Pages: 95-108

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 可微分軌道体上のベクトル場の構造と多項式写像の特異点2004

    • Author(s)
      阿部孝順
    • Journal Title

      京都大学数理解析研究所講究録 1374

      Pages: 95-108

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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