Dynamics of pseudo-Anosov maps and topology of fibered 3-manifolds

• Project/Area Number: 15K04875
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: Kin Eiko 大阪大学, 理学研究科, 准教授 (80378554)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 写像類群 / ３次元双曲多様体 / 擬アノソフ / 位相的エントロピー / 曲線複体 / 群の不変順序 / モノドロミー / 双曲体積 / 組ひも群 / ３次元多様体 / ファイバー / 3次元双曲多様体 / 漸近的移動距離 / ファイバーとモノドロミー / 位相幾何学 / ファイバー多様体 / エントロピー

Outline of Final Research Achievements

We studied two invariants of pseudo-Anosov elements in the mapping class group. One is the　entropy which is the translation length of the pseudo-Anosov element on the Teichmuller space. The other is the asymptotic translation length of the pseudo-Anosov element on the curve complex. We proved that the minimal asymptotic translation length among pseudo-Anosov elements in the hyperelliptic mapping class group of genus g behaves like 1/g^2. (Joint with H. Shin)

We gave a new construction of sequences of pseudo-Anosov braids with small normalized entropies. As an application, we proved that the minimal entropy among pseudo-Anosov elements in the spin mapping class group of genus g behaves like 1/g. Moreover the minimal entropy among pseudo-Anosov skew-palindromic braids with n strands behaves like 1/n. (Joint with S. Hirose)

Research Products

• [Int'l Joint Research] KIAST(韓国)
• [Int'l Joint Research] University of British Columbia(カナダ)
• [Int'l Joint Research] University of British Columbia(カナダ)
• [Int'l Joint Research] KAIST(韓国)
• [Int'l Joint Research] University of British Columbia(カナダ)
• [Journal Article] Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex (2018)
  • Author(s): Eiko Kin, Hyunshik Shin
  • Journal Title: GROUPS, GEOMETRY, AND DYNAMICS Volume: ---
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Braids, orderings and minimal volume cusped hyperbolic 3-manifolds (2018)
  • Author(s): Eiko Kin, Dale Rolfsen
  • Journal Title: GROUPS, GEOMETRY, AND DYNAMICS Volume: ---
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups (2017)
  • Author(s): 金 英子, 廣瀬 進
  • Journal Title: Quarterly Journal of Mathematics Volume: 印刷中
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [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
  • 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
• [Presentation] Small dilatation pseudo-Anosov braids and disk twists (2018)
  • Author(s): 金 英子
  • Organizer: 微分トポロジー18
  • Invited
• [Presentation] Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex (2017)
  • Author(s): 金 英子
  • Organizer: Growth 4
  • Int'l Joint Research / Invited
• [Presentation] Braids, orderings and minimal volume cusped hyperbolic 3-manifolds (2017)
  • Author(s): 金 英子
  • Organizer: The 12th East Asian School of Knots and Related Topics
  • Place of Presentation: 東京大学(東京)
  • Int'l Joint Research / Invited
• [Presentation] Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex (2017)
  • Author(s): 金 英子
  • Organizer: Low dimensional topology and number theory IX
  • Place of Presentation: AiRIMaQ (福岡県・福岡市)
  • Int'l Joint Research / Invited
• [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: 2016-01-26
  • Int'l Joint Research
• [Presentation] Braids, free group automorphisms and orderings (2016)
  • Author(s): 金 英子
  • Organizer: リーマン面に関連する位相幾何学
  • Place of Presentation: 東京大学(東京)
  • Invited
• [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
• [Remarks] Eiko Kin's page
  • URL: http://www.math.sci.osaka-u.ac.jp/~kin/
• [Remarks] 金 英子のホームページ
  • URL: http://www.math.sci.osaka-u.ac.jp/~kin/
• [Remarks] Eiko Kin のホームページ
  • URL: http://www.math.sci.osaka-u.ac.jp/~kin/