Heegaard structures and geometric structures of 3-manifolds

2009 Fiscal Year Final Research Report

• Project/Area Number: 18340018
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Hiroshima University (2007-2009); Osaka University (2006)
• Principal Investigator: SAKUMA Makoto 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)
• 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
• Keywords: 3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

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.

Research Products

• [Journal Article] On hyperbolic once-punctured torus groups: Comparing two tessellations of the complex plane (2010)
  • Author(s): Makoto Sakuma, Warren Dicks
  • Journal Title: Topology and its applications 156 Pages: 1873-1899
  • Peer Reviewed
• [Journal Article] On the distance between two Seifert surfaces of a knot (2009)
  • Author(s): Makoto Sakuma, Kenneth Shackleton
  • Journal Title: Osaka J. Math 46 Pages: 203-221
  • Peer Reviewed
• [Journal Article] Epimorphisms among 2-bridge knot groups, The Zieschang Gedenkschrift (2008)
  • Author(s): Tomotada Ohtsuki, Makoto Sakuma
  • Journal Title: Geometry and Topology Monograph 14 Pages: 417-450
  • Peer Reviewed
• [Journal Article] Epimorphisms among 2-bridge knot groups from the view point of markoff maps, Intelligence of low dimensional topology 2006 (2007)
  • Author(s): Makoto Sakuma
  • Journal Title: World Scientific ( J. Scott carter, etal) Pages: 279-286
  • 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" (2006)
  • Author(s): Hirotaka Akiyoshi, Hideki Miyachi, Makoto Sakuma
  • Journal Title: London Math. Soc. Lecture Note Series(Y. Minsky, etal. ) 329 Pages: 151-185
  • Peer Reviewed
• [Presentation] Epimorphisms between 2-bridge link groups: simple loops on 2-bridge spheres (2010)
  • 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
• [Presentation] Epimorphisms among 2-bridge knot groups and end invariants of SL(2, C)-representations (2009)
  • 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
• [Presentation] Comparing two tessellations associated with punctured torus bundles overs the circle (2008)
  • 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
• [Presentation] On the diatance between two Seifert surfaces of a knot (2007)
  • 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
• [Presentation] n the diatance between two Seifert surfaces of a knot (2007)
  • 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
• [Presentation] Punctured torus groups and 2-bridge knot groups (2006)
  • 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
• [Book] Punctured torus groups and 2-bridge knot groups, Lecture Notes in Math. 1909 (2007)
  • Author(s): Hirotaka Akiyoshi, Makoto Sakuma, Masaaki Wada, Yasushi Yamashita
  • Total Pages: xlvi+252
  • Publisher: Springer, Berline
• [Book] The spaces of Kleinian groups and hyperbolic 3-manifolds (2006)
  • Author(s): Yaiir Minsky, Makoto Sakuma, Caroline Series (eds)
  • Total Pages: vii+390
  • Publisher: London Math. Soc. Lecture Notes Series