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The geometry of character variety and classification of arithmetic Kleinian groups

Research Project

Project/Area Number 20K03612
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionNara Women's University

Principal Investigator

Yamashita Yasushi  奈良女子大学, 自然科学系, 教授 (70239987)

Project Period (FY) 2020-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2022: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Keywords双曲幾何学 / クライン群 / 低次元トポロジー
Outline of Research at the Start

現代の幾何学が主要な対象とする図形に「多様体」と呼ばれるものがある。例えばドーナツの表面(浮き袋)は多様体の例であるが、ドーナツの太さや1周したときの長さなどを変化させることにより様々な形を取りうる。可能な幾何学的形状を全て理解することは重要であり、本研究では解析学・位相幾何学・整数論などの手法を組み合わせたアプローチにより、この問題に取り組む。

Outline of Final Research Achievements

Hyperbolic geometry is an important geometric structure for manifolds of dimension 2 and 3. To understand this structure, we studied index manifolds of the fundamental group of 2-dimensional manifolds. In particular, we studied algebraically and geometrically interesting Klein groups generated by two elements of SL(2,C), which are called arithmetic Klein groups when the number of degrees of the generators is finite.
As an approach to the research, we adopted a method that uses computer experiments in particular, and succeeded in effectively dealing with a problem that was difficult to deal with by conventional methods.

Academic Significance and Societal Importance of the Research Achievements

現代の位相幾何学における主要な研究対象である図形に多様体とよばれるものがあり、それらがどのような形の変形を許容するのかという問題にアプローチすることは、数学の研究を進める上で基本的な意義がある。さらに、算術性などの代数的な手法や写像類群の作用という力学系との関係を明らかにすることにより、分野間の新たなつながりの解明に貢献した。また、本研究は手法としては計算機実験を特徴としており、計算機の応用領域を数理科学に拡げるという形での意義もある。

Report

(4 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (4 results)

All 2022 2021 2020

All Journal Article (1 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 1 results) Presentation (3 results) (of which Invited: 2 results)

  • [Journal Article] Random Kleinian Groups, II Two Parabolic Generators2020

    • Author(s)
      Martin Gaven、O’Brien Graeme、Yamashita Yasushi
    • Journal Title

      Experimental Mathematics

      Volume: 29 Issue: 4 Pages: 443-451

    • DOI

      10.1080/10586458.2018.1477079

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Revisiting the moduli space of right-angled hyperbolic pentagon2022

    • Author(s)
      山下靖
    • Organizer
      Geometry in Low dimensions 2022
    • Related Report
      2022 Annual Research Report
  • [Presentation] Riley sliceと仲間たち2022

    • Author(s)
      山下靖
    • Organizer
      早稲田大学双曲幾何幾何学的群論セミナー
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Computer experiments on Mobius transformations and random Kleinian groups2021

    • Author(s)
      山下靖
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2021 Research-status Report
    • Invited

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Published: 2020-04-28   Modified: 2024-01-30  

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