Heegaard structures and geometric structures of 3-manifolds

2008 Fiscal Year Self-evaluation 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: MAKOTO Sakuma Hiroshima University, 大学院・理学研究科, 教授 (30178602)
• Project Period (FY): 2006 – 2009
• Keywords: 3次元多様体 / へガード分解 / 幾何構造 / 双曲構造 / 穴あきトーラス

Research Abstract

3次元多様体のヘガード構造と幾何構造を結びつける理論を確立するのが, 本研究の長期目標である.この目標達成のために次の研究計画を立てている.
(1) 曲面群のPSL(2, C)表現空間の中で離散表現全体が作る部分空間は, 大雑把に言って, 擬フックス群が作る「大陸」と有限体積双曲多様体からなる「孤島」を「名所」として持つ.その大陸と孤島を結ぶ(錐多様体の連続族が作る)自然な航路を発見すること.
(2) 計画(1)達成のために, (離散的とは限らない)曲面群のPSL(2, C)表現の個性を理解すること

Research Products

• [Journal Article] On the distance between two Seifert surfaces of a knot (2009)
  • Author(s): Makoto Sakuma and Kenneth Shackleton
  • Journal Title: Osaka J. Math 46 Pages: 203-221
• [Journal Article] Epimorphisms among 2-bridge knot groups, The Zieschang Gedenkschrift (2008)
  • Author(s): Tomotada Ohtsuki and Makoto Sakuma
  • Journal Title: Geometry and Topology Monograph 14 Pages: 417-450
• [Journal Article] Epimorphisms among 2-bridge knot groups from the view point of markoff maps (2007)
  • Author(s): Makoto Sakuma
  • Journal Title: Intelligence of low dimensional topology 2006, Eds. J. Scott carter etal (World Scientific.) Pages: 279-286
• [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. Soc (2006)
  • Author(s): Hirotaka Akiyoshi, Hideki MiyachiM and Makoto Sakuma
  • Journal Title: Lecture Note Series 329 Pages: 151-185.
• [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
  • Year and Date: 20090300
• [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
  • Year and Date: 20070900
• [Presentation] On 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
  • Year and Date: 20070600
• [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: Warwick University
  • Year and Date: 20060900
• [Book] Punctured torus groups and 2-bridge knot groups, Lecture Notes in Math. 1909 (2007)
  • Author(s): Hirotaka Akiyoshi, Makoto Sakuma, Masaaki Wada and Yasushi Yamashita
  • Total Pages: xlvi+252
  • Publisher: Springer, Berline
• [Book] London Math. Soc. Lecture Notes Series (2006)
  • Author(s): Yaiir Minsky, Makoto Sakuma and Caroline Series(eds)
  • Total Pages: 329
  • Publisher: The spaces of Kleinian groups and hyperbolic 3-manifolds