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Translation surfacesのモジュライ空間の位相幾何学

Research Project

Project/Area Number 19654009
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionOsaka University

Principal Investigator

大鹿 健一  Osaka University, 大学院・理学研究科, 教授 (70183225)

Co-Investigator(Kenkyū-buntansha) 盛田 健彦  広島大学, 大学院・理学研究科, 教授 (00192782)
高山 晴子  九州大学, 数理学研究院, 助教 (90274430)
Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2009: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2008: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2007: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsTranslation surface / モジュライ空間 / クライン / stratification / タイヒミューラー空間 / translation surface / ビリアード
Research Abstract

今年度はTranslation surfaceの幾何学的研究で得た成果をクライン群の変形空間の研究に応用した.
JMSJの論文では,Bers-Sullivan-Thursonの稠密性予想のparabolicがない場合について,収束性を示す部分を解決法を発表した.一般の予想の解決は投稿中の次の論文で完成させている.
宮地秀樹との共同研究では,Riley sliceに属するKlein群に対して,対応する双曲多様体のuniform modelを構築することに成功した.compressibleなendを持つようなKlein群に対するuniform modelの構成は初めである.これは特にRiley sliceのend付近での多様体の退化の様子をmodelを使って記述できたことによって可能になった.一方で,endから離れた部分では,Minskyらのmodelの構成をbranched coverを使った議論で,compressibleなendを持つ場合に拡張すると,Riley sliceではuniformityが得られることを示した.
WangとLiとの共同研究では通常のKlein群に関するMaskitの組み合わせ定埋を高次元に拡張した.これまでApanasovらによる試みがあったが,そこでは高次元における幾何的有限性の定義が誤って用いられており,全体として議論が完成していなかった.我々の論文では,Bowditchの研究に従い,正しい幾何的有限性の定義を採用し,その下で新たに正しいMaskitの組み合わせ定理の拡張を行った.現在出版されたものは融合積に関する一般化であり,HNN積や,より複雑な状況の組み合わせ定理については,現在同じメンバーで論文を執筆中である.

Report

(3 results)
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (14 results)

All 2010 2009 2008 2007

All Journal Article (4 results) (of which Peer Reviewed: 4 results) Presentation (10 results)

  • [Journal Article] Uniform model for the closure of the Riley slice2010

    • Author(s)
      Ken'ichi Ohshika, Hideki Miyachi
    • Journal Title

      Contemporary Mathematics 510(In press)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Constructing geometrically infinite groups on boundaries of deformation spaces2009

    • Author(s)
      Ken'ichi Ohshika
    • Journal Title

      Journal of Mathematical Society of Japan 61

      Pages: 1261-1291

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On Klein-Maskit Combination Theorem in space I2009

    • Author(s)
      Liuan Li, Ken'ich Ohshika, Xiantao Wang
    • Journal Title

      Osaka Journal of Mathematics 46

      Pages: 1097-1141

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Meromorphic extensions of a class of zeta functions for two-dimensional billiards without eclipse2007

    • Author(s)
      Morita, Takehiko
    • Journal Title

      Tohoku Math. J. 59

      Pages: 167-202

    • NAID

      110006276419

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Algebraic limits viewed through geometric limits2009

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      KMS-AMS joint meeting
    • Place of Presentation
      梨花女子大,ソウル(韓国)
    • Year and Date
      2009-12-17
    • Related Report
      2009 Annual Research Report
  • [Presentation] Applications of geometric limits2009

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      Workshop on Geometric Structures and Geometric Group Theory
    • Place of Presentation
      KIAS,ソウル(韓国)
    • Year and Date
      2009-12-14
    • Related Report
      2009 Annual Research Report
  • [Presentation] Classification of geometric limits and its applications2009

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      Rolf Nevanlinna Colloquium
    • Place of Presentation
      京都大学
    • Year and Date
      2009-09-17
    • Related Report
      2009 Annual Research Report
  • [Presentation] Geometric limits of freely indecomposable Kleinian groups2009

    • Author(s)
      大鹿健一
    • Organizer
      リーマン面・不連続群論研究集会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-01-12
    • Related Report
      2008 Annual Research Report
  • [Presentation] Classification of geometric limits2008

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      Hyperbolic geometry and related topics
    • Place of Presentation
      KIAS, Seoul, Korea
    • Year and Date
      2008-12-15
    • Related Report
      2008 Annual Research Report
  • [Presentation] Deformation spaces of Kleinian groups2008

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      Algebraic Topology, Braids and Mapping Class Groups
    • Place of Presentation
      IMS, National University of Singapore
    • Year and Date
      2008-12-05
    • Related Report
      2008 Annual Research Report
  • [Presentation] Geometry and topology of geometric limits2008

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      リーマン面・不連続群研究集会
    • Place of Presentation
      岡山大学
    • Year and Date
      2008-01-15
    • Related Report
      2007 Annual Research Report
  • [Presentation] Geometric limits viewed through model manifolds2007

    • Author(s)
      大鹿健一
    • Organizer
      離散群と双曲空間の解析学とトポロジー
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-12-10
    • Related Report
      2007 Annual Research Report
  • [Presentation] Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces2007

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      Workshop on Hyperbolic structures on 3-manifolds and large scale geometry of Teichmuller space
    • Place of Presentation
      University of Warwick
    • Year and Date
      2007-07-16
    • Related Report
      2007 Annual Research Report
  • [Presentation] マルコフ系のゼータ関数2007

    • Author(s)
      盛田健彦
    • Organizer
      力学系のゼータ関数-古典力学と量子力学のカオス-
    • Place of Presentation
      中央大学
    • Year and Date
      2007-05-18
    • Related Report
      2007 Annual Research Report

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Published: 2007-04-01   Modified: 2016-04-21  

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