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Research on Schottky groups and Schottky spaces

Research Project

Project/Area Number 10640158
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

SATO Hiroki  Shizuoka University, Science, Professor, 理学部, 教授 (40022222)

Co-Investigator(Kenkyū-buntansha) AKUTAGAWA Kazuo  Shizuoka University, Science, Associate Professor, 理学部, 助教授 (80192920)
NAKANISHI Toshihiro  Nagoya University, Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (00172354)
OKUMURA Yoshihide  Shizuoka University, Science, Associate Professor, 理学部, 助教授 (90214080)
KUMURA Hironori  Shizuoka University, Science, Assistant Professor, 理学部, 助手 (30283336)
奥村 善英  静岡大学, 理学部, 助教授 (30260558)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1998: ¥1,900,000 (Direct Cost: ¥1,900,000)
KeywordsSchottky group / Schottky space / Jorgensen group / Jorgensen's inequality / Kleinian group / Teichmuller space / hyperbolic space / Hausdorff dimension / シヨツトキイ群 / シヨツトキイ空間 / 調和写像
Research Abstract

We have studied the following three themes from 1998 to 1999.
1. Classical Schottky space of real type.
2. Jorgensen numbers.
3. Jorgensen groups.
1. Classical Schottky space of real type. We classified the classical Schottky space of real type of genus two into eight types. We represented the shape of their spaces and found generators for the Schottky modular groups acting on the above spaces. Furthermore we determine fundamental regions for the Schottky modular groups.
2. Jorgensen numbers. We found the best lower bounds for all kinds of the classical Schottky spaces of real type considered in 1. We submited these results to J. Math. Soc. Japan.
3. Jorgensen groups. A Jorgensen group is a non-elementary discrete group whose Jorgensen number is one. In 1998 we considered two one-parameter families of Jorgensen groups. We talked about these results at State University of New York at Stony Brook, Rutgers University at Newark and at New Brunswick in 1998. The results has been printed in Contemporary Mathematics (the proceeding of The Second Ahlfors-Bers Colloquium) in February, 2000. In 1999 we considered distribution of Jorgensen groups on the fiber over the unit circle. In particular, we studied new two one-paramter families of Jorgensen groups on it. We talked about the results at the international conference (ISAAC) in August, 1999. We invited Professor I. Kra at State University of New York at Stony Brook to talk about Schottky groups and the Schottky space in January, 2000. We are preparing to submit some results obtained between April, 1999 and February, 2000.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] Hiroki Sato: "Jorgensen's inequality for classical Schottky groups of real type"J.Math.Soc.Japan. 50. 945-968 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroki Sato: "One-parameter families of extreme discrete groups for Jorgensen's inequality"Contemporary Math,. (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Marjatta Naatanen: "Weil-Peterrsson areas of the moduli space of tori"Results in Math.. 33. 120-133 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Marjatta Naatanen: "Arithmetic Fuchsian groups of signature(0;e_1,e_2,e_3,e_4)with 2 【less than or equal】e_1【less than or equal】e_2【less than or equal】e_3,e_4=∞"Contemporary Math. 240. 269-277 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kazuo Akutagawa: "Spin^c geometry,the Seiberg-Witten equations and Yamabe invariants of Kahler surfaces"Interdisciplinary Inform.Sci.. 5. 55-72 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Reiko Aiyama: "A global correspondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2"Proc. Amer.Math. Soc.. 128. 939-941 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen's inequality for classical Schottky groups of real type"J. Math. Soc. Japan. 50. 945-968 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroki Sato: "One-parameter families of extreme discrete groups for Jorgensen's inequality"Contemporary Math.. (to appear). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Marjatta Naatanen: "Weil-Petersson areas of the moduli space of tori"Results in Math.. 33. 120-133 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Marjatta Naatanen: "Arthmetic Fuchsian groups of signature (0; eィイD21ィエD2, eィイD22ィエD2, eィイD23ィエD2, eィイD24ィエD2 with 2【less than or equal】eィイD21ィエD2【less than or equal】 eィイD22ィエD2【less than or equal】eィイD23ィエD2, eィイD24ィエD2=∞"Contemporary Math.. 240. 269-177 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kazuo Akutagawa: "SpinィイD1CィエD1 geometry, the Seiberg-Witten equations and Yamabe invariants of Kahler surfaces"Interdisciplinary Inform. Sci.. 5. 55-72 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Reiko Aiyama: "A global correspondence between CMC-surfaces in SィイD13ィエD1 and pairs of non-conformal harmonic maps into SィイD12ィエD1"Proc. Amer. Math. Soc.. 128. 939-941 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jφrgensen's inequality for classical Schottky groups of real type"J.Math.Soc.Japan. 50. 945-968 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroki Sato: "One-parameter families of extreme discrete groups for Jorgensen's inequality"Contemporary Math.. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Marjatta Naatanen: "Weil-Petersson areas of the moduli space of tori"Results in Math.. 33. 120-133 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Marjatta Naatanen: "Arithmetic Fuchsian groups of signature(0;e_1,e_2,e_3,e_4)with2【less than or equal】e_1【less than or equal】e_2【less than or equal】e_3,e_4=∞"Contemporay Math.. 240. 269-277 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Kazuo Akutagawa: "Spin^c geometry,the Seiberg-Witten equations and Yamabe invariants of Kahler surfaces"Interdisciplinary Inform.Sci. 5. 55-72 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Reiko Aiyama: "A global correspondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2"Proc.Amer.Math.Soc.. 128. 939-941 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroki Sato: "Jorgensen's inequality for classical Schottky groups of real type" J.Math.Soc.Japan. 50. 945-968 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiroki Sato: "Limits of Schottky groups and the boundary of Teichmuller space" RIMS Koukyuroku. 1065. 51-60 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiroki Sato: "One-parameter families of Jorgensen groups" RIMS Koukyuroku. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kazuo Akutagawa: "Kenmotsu-Bryant type representation formula for constant mean curvature spacelike surfaces in H^3_1(-C^2)" Diff.Geom. its Appl.9. 251-272 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kazuo Akutagawa: "Kenmotsu-Bryant type representation formulas for constant mean curvature surfaces in H^3_1(-C^2) and S^3_1(C^2)" Ann.Global Anal.Geom.17. 49-75 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kazuo Akutagawa: "the Seiberg-Witten equations and Yamabe invariants of kahler surfaces" Interd. Inform.Sci.(1999)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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