Research on symmetries and mapping class groups on the surfaces in low-dimensional manifolds

• Project/Area Number: 16K05156
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: Hirose Susumu 東京理科大学, 理工学部数学科, 教授 (10264144)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 低次元トポロジー / 写像類群 / リーマン面 / 擬アノソフ同相写像 / 結び目 / 双曲多様体 / 分岐被覆空間 / 曲面結び目 / 周期的写像 / 代数曲線 / 3次元ハンドル体 / トポロジー / ３次元ハンドル体

Outline of Final Research Achievements

We made several researches mainly on symmetries and mapping class groups on the surfaces embedded in 3 or 4-dimensional manifolds from the topological viewpoint. Especially, we obtained results on Dehn-twist presentations for the finite group actions on orientable closed surface of genus 3 or 4, Goeritz groups of bridge surfaces of knots in a 3-sphere, and branched virtual fibrations of 3-manifolds.

Academic Significance and Societal Importance of the Research Achievements

局所的に n 次元のユークリッド空間と同一視できる空間のことを n 次元多様体と呼ぶ．この空間を理解することは我々の住んでいる空間を理解するうえで欠かせないものであり，20世紀の後半には，次元 n が大きい場合について明解な理論が作られた．近年，特に n=3,4の場合の研究が急速に進展しており，本研究は，特に 2 次元多様体の上の写像に関する研究を通じて 3, 4 次元多様体の研究を推し進めるものである．

Research Products

• [Journal Article] Volumes of fibered 2-fold branched covers of 3-manifolds (2023)
  • Author(s): Susumu Hirose, Efstratia Kalfagianni, Eiko Kin
  • Journal Title: Journal of Topology and Analysis Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Goeritz groups of bridge decompositions, (2021)
  • Author(s): Susumu Hirose, Daiki Iguchi, Eiko Kin, Yuya Koda
  • Journal Title: International Mathematics Research Notices, Volume: - Issue: 12 Pages: 9308-9356
  • DOI: 10.1093/imrn/rnab001
  • Peer Reviewed / Open Access
• [Journal Article] On hyperbolic surface bundles over the circle as branched double covers of the $3$-sphere (2020)
  • Author(s): Hirose Susumu、Kin Eiko
  • Journal Title: Proceedings of the American Mathematical Society Volume: 148 Issue: 4 Pages: 1805-1814
  • DOI: 10.1090/proc/14825
  • Peer Reviewed / Open Access
• [Journal Article] A construction of pseudo-Anosov braids with small normalized entropies, (2020)
  • Author(s): Susumu Hirose, Eiko Kin,
  • Journal Title: New York Journal of Mathematics Volume: 26 Pages: 562-597
  • Peer Reviewed
• [Journal Article] Dehn twist presentations of finite group actions on oriented closed surface of genus 4 (2019)
  • Author(s): 廣瀬 進
  • Journal Title: 第16回代数曲線論シンポジウム 報告集 Volume: 1 Pages: 125-133
• [Journal Article] Dehn twist presentations of finite group actions on oriented surfaces of genus 3 (2018)
  • Author(s): 廣瀬 進
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 2060 Pages: 118-130
• [Journal Article] Generators for the mapping class group of a nonorientable surface (2018)
  • Author(s): Hirose Susumu
  • Journal Title: Kodai Mathematical Journal Volume: 41 Issue: 1 Pages: 154-159
  • DOI: 10.2996/kmj/1521424829
  • NAID: 130006565544
  • ISSN: 0386-5991, 1881-5472
  • Peer Reviewed
• [Journal Article] The asymptotic behavior of the minimal pseudo-ANOSOV dilatations in the hyperelliptic handlebody groups* (2017)
  • Author(s): Hirose Susumu、Kin Eiko
  • Journal Title: The Quarterly Journal of Mathematics Volume: 68 Issue: 3 Pages: 1035-1069
  • DOI: 10.1093/qmath/hax012
  • Peer Reviewed
• [Journal Article] A uniqueness of periodic maps on surfaces (2016)
  • Author(s): Susumu Hirose, Yasushi Kasahara
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 68 Issue: 4 Pages: 1777-1787
  • DOI: 10.2969/jmsj/06841777
  • NAID: 130005433285
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] 3次元多様体の2重分岐被覆となる曲面束の擬アノソフモノドロミーのエントロピーについて (2022)
  • Author(s): 廣瀬 進
  • Organizer: 研究集会「結び目の数理 V」
• [Presentation] Goeritz groups of bridge decompositions (2020)
  • Author(s): 廣瀬 進
  • Organizer: 2019年度「リーマン面・不連続群論」研究集会
  • Invited
• [Presentation] Goeritz groups of bridge spheres and wicket groups (2019)
  • Author(s): 廣瀬 進
  • Organizer: 研究集会「拡大KOOK セミナー2019」
• [Presentation] Dehn twist presentations of finite group actions on oriented closed surface of genus 4 (2018)
  • Author(s): 廣瀬 進
  • Organizer: 第 16 回代数曲線論シンポジウム
  • Invited
• [Presentation] 結び目の FPB 表示と Arf 不変量について (2018)
  • Author(s): 廣瀬 進
  • Organizer: 研究集会「2017年度琉球結び目セミナー」
• [Presentation] Dehn twist presentations of finite group actions on oriented surfaces of genus 3 (2017)
  • Author(s): 廣瀬 進
  • Organizer: 研究集会「変換群を核とする代数的位相幾何学」
  • Invited
• [Presentation] 結び目の FPB 表示と Arf 不変量について (2017)
  • Author(s): 廣瀬 進
  • Organizer: Knotting Nagoya (結び目の数理セミナー )
  • Invited
• [Presentation] Dehn twist presentations of finite group actions on oriented surfaces of genus 3 (2017)
  • Author(s): 廣瀬 進
  • Organizer: 研究集会「リーマン面に関連する位相幾何学」
  • Invited
• [Presentation] An invitation for the mapping class groups of handlebodies (2017)
  • Author(s): 廣瀬 進
  • Organizer: ハンドル体結び目とその周辺10・Hurwitz action 7
  • Invited
• [Presentation] Symmetric braids arising from monodromies of branched virtual fibrations of closed 3-manifolds (2017)
  • Author(s): 金 英子, 廣瀬 進
  • Organizer: 広島大学 トポロジー・幾何セミナー
  • Invited
• [Presentation] 超楕円的ハンドル体群の有限表示について (2016)
  • Author(s): 廣瀬 進
  • Organizer: 早稲田双曲幾何幾何学的群論セミナー
  • Place of Presentation: 早稲田大学（東京都・新宿区）
  • Invited
• [Presentation] 超楕円的ハンドル体群の有限表示 (2016)
  • Author(s): 廣瀬 進
  • Organizer: 研究集会「リーマン面に関連する位相幾何学」
  • Place of Presentation: 東京大学大学院数理科学研究科（東京都・目黒区）
  • Invited
• [Presentation] クライン４次曲線の自己同型群の Dehn twist 表示について (2016)
  • Author(s): 廣瀬 進
  • Organizer: 第14回代数曲線論シンポジウム
  • Place of Presentation: 神奈川工科大学（神奈川県・横浜市）
  • Invited
• [Book] Low Dimensional Topology and Number Theory (tentative), Springer Proc. Math. Stat. Editors Masanori Morishita, Hiroaki Nakamura, Jun Ueki (2024)
  • Author(s): Susumu Hirose, Eiko Kin
  • Publisher: Springer, Cham