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Study on Teichmuller modular groups through flat structures

Research Project

Project/Area Number 26400151
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionWaseda University

Principal Investigator

KOMORI YOHEI  早稲田大学, 教育・総合科学学術院, 教授 (70264794)

Project Period (FY) 2014-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywordsリーマン面 / コクセター群 / サラム数 / 複素解析幾何学 / 複素解析幾何 / 国際研究者交流
Outline of Final Research Achievements

Following the closed connection between Teichmuller modular subgroups and geometric Coxeter groups,
we constructed pseudo-Anosov mappings from Coxeter systems and considered their growth rates. Our Coxeter systems were obtained from affine Coxeter diagrams by adding one edge. Taking the double coverings of these diagrams, we showed that our Coxeter systems belong to hyperbolic class when the number of vertices of diagrams are less than or equal to 8, while they belong to higher rank class when the number of vertices of diagrams are more than or equal to 10. As an application of our result, we can construct a series of pseudo-Anosov mappings whose growth rates are 2-Salem numbers.

Academic Significance and Societal Importance of the Research Achievements

曲面から曲面自身への自己同相写像の反復合成を曲面上の離散的な時間発展と思うと、曲面の種類及び同相写像の種類により複雑になる。特に種数が2以上の曲面における擬アノソフ写像は、伸びる方向と縮む方向に分解でき、トーラスの場合の双曲変換の類似になっている。この伸縮の割合を擬アノソフ写像の拡大率といい、その数論的性質など詳しく調べられている。
今回の研究では鏡映変換から定まるコクセター系という代数系を用いて、2サラム数という代数的整数を拡大率に持つ擬アノソフ写像を構成した。

Report

(7 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (26 results)

All 2020 2019 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (3 results) Journal Article (9 results) (of which Int'l Joint Research: 2 results,  Open Access: 4 results,  Peer Reviewed: 4 results,  Acknowledgement Compliant: 1 results) Presentation (13 results) (of which Int'l Joint Research: 3 results,  Invited: 9 results) Funded Workshop (1 results)

  • [Int'l Joint Research] フリブール大学(スイス)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] フリブール大学(スイス)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] フリブール大学(スイス)

    • Related Report
      2016 Research-status Report
  • [Journal Article] 有限上半平面グラフについて2020

    • Author(s)
      小森 洋平 ・安井 拓朗
    • Journal Title

      学術研究(自然科学編 )

      Volume: 68 Pages: 51-65

    • NAID

      40022219448

    • Related Report
      2019 Annual Research Report
    • Open Access
  • [Journal Article] On finite upper half plane graphs2019

    • Author(s)
      Yohei Komori
    • Journal Title

      Mathematisches Forschungsinstitut Oberwolfach Report

      Volume: 17 Issue: 2 Pages: 12-13

    • DOI

      10.4171/owr/2019/17

    • NAID

      40022219448

    • Related Report
      2019 Annual Research Report
    • Open Access
  • [Journal Article] 古典幾何における内接五角形の面積公式2019

    • Author(s)
      梅澤 瑠奈・小森 洋平・安井 拓朗
    • Journal Title

      早稲田大学教育総合科学学術院 学術研究(自然科学編)

      Volume: 67 Pages: 11-24

    • NAID

      40021861011

    • Related Report
      2018 Research-status Report
  • [Journal Article] On 3-dimensional hyperbolic Coxeter pyramids2018

    • Author(s)
      Y. Komori and Y. Umemoto
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B66 Pages: 213-230

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] On 3-dimensional hyperbolic Coxeter pyramids2017

    • Author(s)
      Yohei Komori and Yuriko Umemoto
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: -

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] 立方体と分割合同な三角錐の構成について2016

    • Author(s)
      小森洋平、利根川真隆、雪田友成
    • Journal Title

      学術研究 早稲田大学 教育・総合科学学術院

      Volume: 65 Pages: 1-14

    • NAID

      120006230553

    • Related Report
      2016 Research-status Report
    • Open Access
  • [Journal Article] On the growth rate of ideal Coxeter groups in hyperbolic 3-space2015

    • Author(s)
      Yohei Komori and Tomoshige Yukita
    • Journal Title

      Proc. Japan Acad. Ser. A Math. Sci.

      Volume: Volume 91, Number 10 Issue: 10 Pages: 155-159

    • DOI

      10.3792/pjaa.91.155

    • NAID

      40020687058

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] POLYHEDRAL REALIZATION OF A THURSTON COMPACTIFICATION2014

    • Author(s)
      M. Gendulphe and Y. Komori
    • Journal Title

      Ann. Fac. Sci. Toulouse, Math.

      Volume: 10 Issue: 13 Pages: 1-15

    • DOI

      10.5802/afst.1398

    • URL

      https://ocu-omu.repo.nii.ac.jp/records/2016743

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] 射影図を用いた双曲コクセター理想多面体の分類2014

    • Author(s)
      小森洋平・雪田友成
    • Journal Title

      学術研究

      Volume: 63 Pages: 47-56

    • NAID

      120005596574

    • Related Report
      2014 Research-status Report
  • [Presentation] On finite upper half plane graphs2019

    • Author(s)
      Yohei Komori
    • Organizer
      Mini-Workshop: Reflection Groups in Negative Curvature
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] タイヒミュラー空間の角度パラメータについて2019

    • Author(s)
      小森洋平
    • Organizer
      タイヒミュラー空間と双曲幾何学
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Growth of hyperbolic Coxeter groups2018

    • Author(s)
      Yohei Komori
    • Organizer
      Growth in Topology and Number Theory: Volumes, Entropy, and L2-torsion
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Construction of pseudo-Anosov automorphisms whose dilatations are 2-Salem numbers2018

    • Author(s)
      小森洋平
    • Organizer
      日本数学会
    • Related Report
      2017 Research-status Report
  • [Presentation] On spectral radii of Coxeter elements for some bipartite Coxeter diagrams2017

    • Author(s)
      Yohei Komori
    • Organizer
      Geometry seminar at University of Fribourg
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Growth functions of hyperbolic groups2017

    • Author(s)
      Yohei Komori
    • Organizer
      colloquium talk at University of Fribourg
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On Schwarz automorphic functions2016

    • Author(s)
      Yohei Komori
    • Organizer
      Topology and Analysis of Discrete Groups and Hyperbolic Spaces
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 3次元双曲理想コクセター多面体の増大度について2015

    • Author(s)
      小森洋平
    • Organizer
      日本数学会2015年度秋季総合分科会
    • Place of Presentation
      京都産業大学
    • Year and Date
      2015-09-14
    • Related Report
      2015 Research-status Report
  • [Presentation] Coxeter garlands と 2-Salem 数2015

    • Author(s)
      小森洋平
    • Organizer
      日本数学会2015年度年会幾何学分科会一般講演
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-21
    • Related Report
      2014 Research-status Report
  • [Presentation] Projective embeddings of the Teichmuller spaces2015

    • Author(s)
      Yohei Komori
    • Organizer
      「リーマン面・不連続群論」研究集会
    • Place of Presentation
      大阪大学中之島センター
    • Year and Date
      2015-02-15
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Arithmetic aspects of growth rates for hyperbolic Coxeter groups2015

    • Author(s)
      Yohei Komori
    • Organizer
      Complex Hyperbolic Geometry and Related Topics
    • Place of Presentation
      岡山理科大
    • Year and Date
      2015-01-10
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] トーラス上のリーマン面の退化族について2014

    • Author(s)
      小森洋平
    • 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
    • Year and Date
      2014-06-27
    • Related Report
      2014 Research-status Report
  • [Presentation] トーラス上のリーマン面の退化族について2014

    • Author(s)
      小森洋平
    • Organizer
      学習院大学トポロジーセミナー
    • Place of Presentation
      学習院大学
    • Year and Date
      2014-04-30
    • Related Report
      2014 Research-status Report
    • Invited
  • [Funded Workshop] Growth 32016

    • Place of Presentation
      早稲田大学教育学部
    • Related Report
      2016 Research-status Report

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Published: 2014-04-04   Modified: 2021-02-19  

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