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Dynamics of modular groups on infinite dimensional Teichmuller spaces

Research Project

Project/Area Number 14540156
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOchanomizu University

Principal Investigator

MATSUZAKI Katsuhiko  Ochanomizu Univ., Faculty of Science, Associate Prof., 理学部, 助教授 (80222298)

Co-Investigator(Kenkyū-buntansha) SUGAWA Toshiyuki  Hiroshima Univ., Grad.School of Science, Associate Prof., 大学院・理学研究科, 助教授 (30235858)
Project Period (FY) 2002 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2003: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2002: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsTeichmuller space / modular group / Riemann surface / quasiconformal map / hyperbolic geometry / Schwarzian derivative / Bers embedding / univalent function / モジュライ空間 / 双曲幾何
Research Abstract

Teichmueller spaces are not homogeneous spaces and their mudular groups do not act transitively. For compact Riemann surfaces, modular groups act discontinuously, but this is not the case for infinite dimensional Teichmueller spaces. We study the moduli spaces of Riemann surafces of infinite type by considering the chaotic behavior of the action of modular groups. For a viewpoint of general topology, the moduli space is either metrizable or not of the first separation axiom. However, except for a singular part, it can possess a certain geometric structure. In this research, we characterize this stable region by hyperbolic geometric structure of a Riemann surface and construct a contracted moduli space by the completion of the stable region. Consequently, we can describe the closure of a point set in terms of the geomery of Riemann surfaces, which is a point of teh contracted module space.
We considered the space of pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian model of the universal Teichmueller space and classical families of univalent functions. We also investigated geometric properties of univalent functions with a prescribed growth of the Schwarzian derivative and found that they are starlike or convex according to the distance to the origin in the Bers embedding of the universal Teichmueller space.

Report

(3 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] K.Matsuzaki: "The infinite direct product of Dehntwists acting on infinite dimensional Teichmuller spaces."Kodai Math.J.. 26. 279-287 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Matsuzaki: "Inclusion relations between the Bers embeddings of Teichmuller spaces"Israel J.Math.. 140. 113-124 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Matsuzaki: "A countable Teichmuller modular group"Trans.Amer.Math.Soc.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Sugawa: "A remark on the Ahlfors-Lehto univalence criterion"Ann.Acad.Sci.Fenn.. 27. 151-161 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Sugawa: "Estimates of hyperbolic metric with applications to Teichmuller spaces"Kyungpook Math.J.. 42. 51-60 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Komori, T.Sugawa: "Bers embedding of the Teichmuller space of a once-punctured tours"Conform.Geom.Dyn.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Matauzaki: "The infinite direct product of Dehn twists acting on infinite dimensional Teichmuller spaces"Kodai Math. J.. 26. 279-287 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Matauzaki: "Inclusion relations between the Bers embeddings of Teichmuller spaces"Israel J.Math.. 140. 113-124 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Matauzaki: "A countable Teichmuller modular group"(in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Sugawa: "A remark on the Ahlfors-Lehto univalence criterion"Ann.Acad.Sci.Fenn.. 27. 151-161 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Sugawa: "Estimates of hyperbolic metric with applications to Teichmuller spaces"Kyungpook Math.J.. 42. 51-60 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Komori, T.Sugawa: "Bers embedding of the Teichmuller space of a once-punctured torus"(in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Matsuzaki: "Conservative action of Kieinian groups with respect to the Patterson-Sullivan measure"Comput.Methods Funct.Theory. 2. 469-479 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Matsuzaki: "The infinite direct product of Dehn twists acting on infinite dimensional Teichrmuller spaces"Kodai Math J.. 26. 279-287 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Matsuzaki: "An extension of the collar lemma"数理解析研究所講究録. 1329. 58-61 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Matsuzaki: "A countable Teichmuller modular group"Trans.Amer.Math.Soc.. 発表予定(未定).

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Sugawa: "Inner radius of univalence for a strongly starlike domain"Monatsh.Math. 139. 61-68 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Sugawa: "Uniformly perfect sets : analytic and geometric aspects"Sugaku Expo.. 16. 225-242 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Matsuzaki: "Simply connected domains on a hyperbolic surface"New Zealand J.Math.. 31. 159-164 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Matsuzaki: "Inclusion relations between the Bers embeddings of Teichmuller spaces"Israel J.Math.. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Matsuzaki: "Indecomposable continua and the limit sets of Kleinian groups"Contemporary Math.. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Matsuzaki: "The action of isotropy subgroups of the modular groups on infinite dimensional Teichmuller spaces"数理解析研究所講究録. 1270. 84-87 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Sugawa: "Estimates of hyperbolic metric with applications to Teichmuller spaces"Kyungpook Math.J.. 42. 51-60 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Sugawa: "A remark on the Ahlfors-Lehto univalence criterion"Ann.Acad.Sci.Fenn.. 27. 151-162 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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