A study of Entropies of pseudo-Anosov mapping classes and hyperbolic fibered 3-manifolds

• Project/Area Number: 24740039
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: Kin Eiko 大阪大学, 理学(系)研究科(研究院), 准教授 (80378554)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 位相幾何 / 双曲多様体 / ファイバー多様体 / 写像類群 / 擬アノソフ / エントロピー / 群の不変順序 / トポロジー / 幾何学 / 3次元双曲多様体 / トレイントラック / タイヒミュラー多項式 / train track

Outline of Final Research Achievements

We consider the mapping class group of an orientable surface. Given a mapping class, we can build its mapping torus which is a fibered 3-manifold. It is known by Thurston that a mapping class is　pseudo-Anosov (pA) if and only if its mapping torus is hyperbolic. If we fix a fibered 3-manifold M with the first Betti number greater than 1, there are infinitely many ways to represent M as mapping tori. I consider the so-called magic 3-manifold N which is a hyperbolic 3-manifold with 3 cusps.
I explicitly construct every pA mapping class whose mapping torus is homeomorphic N. By using these pA mapping classes, I study some properties on bi-orderings of the fundamental groups of some hyperbolic fibered 3-manifolds.

Research Products

• [Int'l Joint Research] University of British Columbia(カナダ)
• [Journal Article] The boundary of a fibered face of the magic 3-manifold and the asymptotic behavior of the minimal pseudo-Anosovs dilatations (2016)
  • Author(s): Eiko Kin, Mitsuhiko Takasawa
  • Journal Title: Hiroshima Mathematical Journal Volume: 印刷中
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Dynamics of the monodromies of the fibrations on the magic 3-manifold (2015)
  • Author(s): Eiko Kin
  • Journal Title: New York Journal of Mathematics Volume: 21 Pages: 547-599
  • Peer Reviewed / Open Access
• [Journal Article] Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior (2013)
  • Author(s): Eiko Kin, Sadayoshi Kojima and Mitsuhiko Takasawa
  • Journal Title: Algebraic and geometric topology Volume: 未定
  • Peer Reviewed
• [Journal Article] Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior (2013)
  • Author(s): Eiko Kin and Mitsuhiko Takasawa
  • Journal Title: Jounal of the Mathematical Society of Japan Volume: 未定
  • NAID: 10031177287
  • Peer Reviewed
• [Journal Article] Notes on pseudo-Anosovs with small dilatations coming from the magic 3-manifold (2013)
  • Author(s): Eiko Kin
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 未定
  • Peer Reviewed
• [Presentation] Braids, free group automorphisms and orderings (2016)
  • Author(s): 金 英子
  • Organizer: リーマン面・不連続群論研究集会
  • Place of Presentation: 東京工業大学 (東京)
  • Year and Date: 2016-02-13
  • Invited
• [Presentation] The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups (2016)
  • Author(s): 金 英子
  • Organizer: The 11th East Asian School of Knots and Related Topics
  • Place of Presentation: 大阪市立大学 (大阪)
  • Year and Date: 2016-01-16
  • Int'l Joint Research
• [Presentation] Braids, automorphisms and orderings (2015)
  • Author(s): 金 英子
  • Organizer: PIMS (Pacific Institute for the mathematical sciences) topology seminar
  • Place of Presentation: University of British Columbia (バンクーバー・カナダ)
  • Year and Date: 2015-11-04
  • Int'l Joint Research / Invited
• [Presentation] The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups (2015)
  • Author(s): 金 英子
  • Organizer: 広島大学幾何トポロジーセミナー
  • Place of Presentation: 広島大学 (広島)
  • Year and Date: 2015-07-21
  • Invited
• [Presentation] Pseudo-Anosov elements with small dilatations in the spherical wicket braid groups (2015)
  • Author(s): 金 英子
  • Organizer: 冬の力学系研究集会
  • Place of Presentation: 日本大学軽井沢研究所(長野県)
  • Year and Date: 2015-01-12
• [Presentation] The asymptotic behavior of minimal dilatations in the spherical wicket braid groups (2015)
  • Author(s): 金 英子
  • Organizer: Hurwitz action~HINERU~
  • Place of Presentation: 大阪電気通信大学(大阪)
  • Year and Date: 2015-01-10
  • Invited
• [Presentation] The magic 3-manifold, horseshoe braids and minimal dilatation problem, (2014)
  • Author(s): 金 英子
  • Organizer: Boston University/Keio University Workshop 2014, Dynamical Systems
  • Place of Presentation: Boston University (USA)
  • Year and Date: 2014-09-19
  • Invited
• [Presentation] Dynamics of the monodromies of the fibrations on the magic 3-manifold (2014)
  • Author(s): 金 英子
  • Organizer: Geometric Group Theory and Geometric Structures
  • Place of Presentation: KAIST (Korea)
  • Year and Date: 2014-08-07
• [Presentation] Monodromies of fibrations on the magic 3-manifold (2014)
  • Author(s): 金 英子
  • Organizer: 冬の力学系研究集会
  • Place of Presentation: 広島大学
• [Presentation] Minimal dilatations of pseudo-Anosovs and a mystery of the magic $3$-manifold (2013)
  • Author(s): 金 英子
  • Organizer: Topology, Geometry and Group Theory, Informed by Experiment
  • Place of Presentation: Brown University
• [Presentation] Monodromies of fibrations on the magic 3-manifold (2013)
  • Author(s): 金 英子
  • Organizer: トポロジーとコンピュータ2013
  • Place of Presentation: 明治大学
• [Presentation] Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior
  • Author(s): 金 英子
  • Organizer: リーマン面・不連続群論」研究集会
  • Place of Presentation: 大阪大学
  • Invited
• [Remarks] Eiko Kin のホームページ
  • URL: http://www.math.sci.osaka-u.ac.jp/~kin/
• [Remarks] http://www.math.sci.osaka-u.ac.jp/~kin/