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Heegaard structures and geometric structures of 3-manifolds

Research Project

Project/Area Number 18340018
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionHiroshima University (2007-2009)
Osaka University (2006)

Principal Investigator

SAKUMA Makoto (MAKOTO Sakuma)  Hiroshima University, 大学院・理学研究科, 教授 (30178602)

Co-Investigator(Kenkyū-buntansha) KAMADA Seiichi  広島大学, 大学院・理学研究科, 教授 (60254380)
NAGAI Toshitaka  広島大学, 大学院・理学研究科, 教授 (40112172)
MATSUMOTO Takao  広島大学, 大学院・理学研究科, 名誉教授 (50025467)
秋吉 宏尚  大阪市立大学, 大学院・理学研究科, 特任准教授 (80397611)
和田 昌昭  奈良女子大学, 理学部, 教授 (80192821)
山下 靖  奈良女子大学, 理学部, 准教授 (70239987)
大鹿 健一  大阪大学, 大学院・理学研究科, 教授 (70183225)
梅原 雅顕  大阪大学, 大学院・理学研究科, 教授 (90193945)
Co-Investigator(Renkei-kenkyūsha) UMEHARA Masaaki  大阪大学, 大学院・理学研究科, 教授 (90193945)
OHSHIKA Ken'ichi  大阪大学, 大学院・理学研究科, 教授 (70183225)
KONNO Kazuhiro  大阪大学, 大学院・理学研究科, 教授 (10186869)
MABUCHI Toshiki  大阪大学, 大学院・理学研究科, 教授 (80116102)
WADA Masaaki  大阪大学, 大学院・理学研究科, 教授 (80192821)
MIYACHI Hideki  大阪大学, 大学院・理学研究科, 教授 (40385480)
KOBAYASHI Tsuyoshi  奈良女子大学, 理学部, 教授 (00186751)
YAMASHITA Yasushi  奈良女子大学, 理学部, 教授 (70239987)
MORIMOTO Kanji  甲南大学, 理学部, 教授 (90200443)
NAKANISHI Toshihiro  島根大学, 総合理工学部, 教授 (00172354)
KOMORI Yohei  大阪市立大学, 大学院・理学研究科, 准教授 (70264794)
AKIYOSHI Hirotaka  近畿大学, 理工学部, 准教授 (80397611)
Project Period (FY) 2006 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥9,900,000 (Direct Cost: ¥8,100,000、Indirect Cost: ¥1,800,000)
Fiscal Year 2009: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2008: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2007: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2006: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywords3次元多様体 / ヘガード分解 / 幾何構造 / 双曲構造 / 穴あきトーラス / 曲面束 / Cannon-Thurston map / 標準的分割 / 2橋結び目 / small cancellation theory / へガード分解 / Cannon-Thurston写像 / Canonical decomposition / Cannon-Thurston-Dicks分解 / 穴あきトーラス束 / 強可逆結び目 / 垣水複体 / Montesinos knot / virtual fiber / 擬フックス群 / ザイフェルト曲面
Research Abstract

We have concentrated on the study of the once-punctured torus, the simplest hyperbolic surface, believing that it would bring us to deep understanding of general hyperbolic surfaces, and obtained the following results. (1) We gave a complete description and proof to Jorgensen's theory on quasifuchsian punctured torus groups. (2) We found an intimate relation between the following two tessellations associated with a punctured torus bundles over the circle ; the Cannon-Thurston-Dicks fractal tessellation and the cusp triangulation induced by the canionical decomposition. We also proposed a conjecture concerning the canonical decompositions of punctured surface bundles over the circle. (3) We gave a complete characterization of those essential simple loops on the bridge sphere of a 2-bridge knot which are null-homotopic in the knot complement.

Report

(6 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report   Self-evaluation Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (38 results)

All 2010 2009 2008 2007 2006

All Journal Article (19 results) (of which Peer Reviewed: 9 results) Presentation (13 results) Book (6 results)

  • [Journal Article] On hyperbolic once-punctured torus groups: Comparing two tessellations of the complex plane2010

    • Author(s)
      Makoto Sakuma, Warren Dicks
    • Journal Title

      Topology and its applications 156

      Pages: 1873-1899

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] On hyperbolic once-punctured torus bundles III : Comparing two tessellations of the complex plane2010

    • Author(s)
      Makoto Sakuma
    • Journal Title

      Topology and its Applications (印刷中)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the distance between two Seifert surfaces of a knot2009

    • Author(s)
      Makoto Sakuma, Kenneth Shackleton
    • Journal Title

      Osaka J. Math 46

      Pages: 203-221

    • NAID

      120004843529

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the distance between two Seifert surfaces of a knot2009

    • Author(s)
      Makoto Sakuma
    • Journal Title

      Osaka Journal of Mathematics 46

      Pages: 203-221

    • NAID

      120004843529

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the distance between two Seifert surfaces of a knot2009

    • Author(s)
      Makoto Sakuma and Kenneth Shackleton
    • Journal Title

      Osaka J. Math 46

      Pages: 203-221

    • NAID

      120004843529

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] On the distance of two Seifert surfaces of a knot2009

    • Author(s)
      M. Sakuma, K. Schackleton
    • Journal Title

      Osaka Journal of Mathematics 46

      Pages: 203-221

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Epimorphisms among 2-bridge knot groups, The Zieschang Gedenkschrift2008

    • Author(s)
      Tomotada Ohtsuki, Makoto Sakuma
    • Journal Title

      Geometry and Topology Monograph 14

      Pages: 417-450

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Epimorphisms among 2-bridge knot groups, The Zieschang Gedenkschrift2008

    • Author(s)
      Tomotada Ohtsuki and Makoto Sakuma
    • Journal Title

      Geometry and Topology Monograph 14

      Pages: 417-450

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Epimorphisms between 2-bridge knot groups2008

    • Author(s)
      T. Ohtski, R. Riley, M. Sakuma
    • Journal Title

      Geometry and Topology Monograph 1909(印刷中)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Epimorphisms among 2-bridge knot groups from the view point of markoff maps, Intelligence of low dimensional topology 20062007

    • Author(s)
      Makoto Sakuma
    • Journal Title

      World Scientific ( J. Scott carter, etal)

      Pages: 279-286

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Epimorphisms among 2-bridge knot groups from the view point of markoff maps2007

    • Author(s)
      Makoto Sakuma
    • Journal Title

      Intelligence of low dimensional topology 2006, Eds. J. Scott carter etal (World Scientific.)

      Pages: 279-286

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Punduced torus groups and 2-bridge knot groups (I)2007

    • Author(s)
      H.Akiyoshi, H.sakuma, H.woda, Y.Yamashita
    • Journal Title

      Lecture Notes in Mathematics, Springer-Verlag (印刷中)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Epimorphisms between 2-bridge knot groups2007

    • Author(s)
      T.Ohtsuki, R.Riloy, M.Sakuma
    • Journal Title

      geometry and Topology Monograph (印刷中)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Epimorphisms between 2-bridge knot groups from the view point of markoff maps2007

    • Author(s)
      M.Sakuma
    • Journal Title

      Proc. of the workshop, Intelligence of lowdim. topology (印刷中)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] variations of McShane's identity for punctured surface groups, Proceedings of the workshop "Spaces of Kleinan groups and hyperbolic 3-manifolds"2006

    • Author(s)
      Hirotaka Akiyoshi, Hideki Miyachi, Makoto Sakuma
    • Journal Title

      London Math. Soc. Lecture Note Series(Y. Minsky, etal. ) 329

      Pages: 151-185

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] variations of McShane's identity for punctured surface groups, Proceedings of the workshop "Spaces of Kleinan groups and hyperbolic 3-manifolds", Eds. Y. Minsky etal. London Math. Soc2006

    • Author(s)
      Hirotaka Akiyoshi, Hideki MiyachiM and Makoto Sakuma
    • Journal Title

      Lecture Note Series 329

      Pages: 151-185

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Variations of Mcshanes indentify for punctured surface groups2006

    • Author(s)
      H.Akiyoshi, H.Miyachi, H.Sakuma
    • Journal Title

      London Math. Soc. Lect. Notes Series 329

      Pages: 151-185

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On topologically fame kleinian groups with bounded geometry2006

    • Author(s)
      K.Ohshika, H.Miyachi
    • Journal Title

      London Math. Soc. Lecture Notes Series 329

      Pages: 29-48

    • Related Report
      2006 Annual Research Report
  • [Journal Article] OPTi's algorithm for disaeteness determination2006

    • Author(s)
      M.Wada
    • Journal Title

      Experiment. Math. 15・1

      Pages: 61-66

    • Related Report
      2006 Annual Research Report
  • [Presentation] Epimorphisms between 2-bridge link groups: simple loops on 2-bridge spheres2010

    • Author(s)
      Makoto Sakuma
    • Organizer
      The 6th East Asian School of Knots and Related Topics
    • Place of Presentation
      Nankai University, 中国
    • Year and Date
      2010-01-27
    • Related Report
      2009 Final Research Report
  • [Presentation] Epimorphisms between 2-bridge link groups : simple loops on 2-bridge spheres2010

    • Author(s)
      Makoto Sakuma
    • Organizer
      The 6^<th> East Asian School of Knots and Related Topics
    • Place of Presentation
      Nankai University
    • Year and Date
      2010-01-27
    • Related Report
      2009 Annual Research Report
  • [Presentation] Epimorphisms among 2-bridge knot groups and end invariants of SL(2, C)-representations2009

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際研究集会「Swiss Knots」
    • Place of Presentation
      Univ. Fribourg, スイス
    • Year and Date
      2009-03-19
    • Related Report
      2009 Final Research Report
  • [Presentation] Epimorphisms among 2-bridge knots and end invariants of SL(2, C)2009

    • Author(s)
      作間 誠
    • Organizer
      Swiss Knots
    • Place of Presentation
      フリブール大学(スイス)
    • Year and Date
      2009-03-19
    • Related Report
      2008 Annual Research Report
  • [Presentation] Epimorphisms among 2-bridge knot groups and end invariants of SL(2, C)-representations2009

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際研究集会「Swiss Knots」
    • Place of Presentation
      Univ. Fribourg
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Comparing two tessellations associated with punctured torus bundles overs the circle2008

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際研究集会「Intelligence of low dimensional topology」
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2008-10-08
    • Related Report
      2009 Final Research Report 2008 Self-evaluation Report
  • [Presentation] On the diatance between two Seifert surfaces of a knot2007

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際研究集会「Braids, Groups and Manifolds in Toulouse」
    • Place of Presentation
      Univ. Paul Sabatier, フランス
    • Year and Date
      2007-09-08
    • Related Report
      2009 Final Research Report
  • [Presentation] n the diatance between two Seifert surfaces of a knot2007

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際会議「Geometry and Topology Conference」
    • Place of Presentation
      北京大学, 中国
    • Year and Date
      2007-06-19
    • Related Report
      2009 Final Research Report
  • [Presentation] On the distance between two Seifert surfaces of a knot2007

    • Author(s)
      M. Sakuma
    • Organizer
      Geometric Topology Conference-Beijing 2007
    • Place of Presentation
      北京大学
    • Year and Date
      2007-06-19
    • Related Report
      2007 Annual Research Report
  • [Presentation] On the diatance between two Seifert surfaces of a knot2007

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際研究集会「Braids, Groups and Manifolds in Toulouse」
    • Place of Presentation
      Univ. Paul Sabatier
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] On the diatance between two Seifert surfaces of a knot2007

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際会議「Geometry and Topology Conference」
    • Place of Presentation
      北京大学
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Punctured torus groups and 2-bridge knot groups2006

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際研究集会「Analytic aspects of low dimensional geometry」
    • Place of Presentation
      arwick University, 英国
    • Year and Date
      2006-09-15
    • Related Report
      2009 Final Research Report
  • [Presentation] Punctured torus groups and 2-bridge knot groups2006

    • Author(s)
      Makoto Sakuma
    • Organizer
      国際研究集会「Analytic aspects of low dimensional geometry」
    • Place of Presentation
      Warwick University
    • Related Report
      2008 Self-evaluation Report
  • [Book] Punctured torus groups and 2-bridge knot groups, Lecture Notes in Math. 19092007

    • Author(s)
      Hirotaka Akiyoshi, Makoto Sakuma, Masaaki Wada, Yasushi Yamashita
    • Publisher
      Springer, Berline
    • Related Report
      2009 Final Research Report
  • [Book] Punctured torus groups and 2-bridge knot groups, Lecture Notes in Math. 19092007

    • Author(s)
      Hirotaka Akiyoshi, Makoto Sakuma, Masaaki Wada and Yasushi Yamashita
    • Publisher
      Springer, Berline
    • Related Report
      2008 Self-evaluation Report
  • [Book] Punctured torus groups and 2-bridge knot groups(1), Lecture Notes in Mathematics Vol 19092007

    • Author(s)
      H. Akiyoshi, M. Sakuma, M. Wada, Y. Yamashita
    • Total Pages
      295
    • Publisher
      Springer Verlag
    • Related Report
      2007 Annual Research Report
  • [Book] The spaces of Kleinian groups and hyperbolic 3-manifolds2006

    • Author(s)
      Yaiir Minsky, Makoto Sakuma, Caroline Series (eds)
    • Publisher
      London Math. Soc. Lecture Notes Series
    • Related Report
      2009 Final Research Report
  • [Book] London Math. Soc. Lecture Notes Series2006

    • Author(s)
      Yaiir Minsky, Makoto Sakuma and Caroline Series(eds)
    • Total Pages
      329
    • Publisher
      The spaces of Kleinian groups and hyperbolic 3-manifolds
    • Related Report
      2008 Self-evaluation Report
  • [Book] Spaces of Kleinian groups2006

    • Author(s)
      Y.Minsky, M.Sakuma, C.Series
    • Total Pages
      390
    • Publisher
      Cambridge University Press
    • Related Report
      2006 Annual Research Report

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Published: 2006-04-01   Modified: 2016-04-21  

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