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The asymptotic behavior of the Reidemeister torsion for degenerate hyperbolic structures

Research Project

Project/Area Number 17K05240
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionWaseda University (2022)
Akita University (2017-2021)

Principal Investigator

YAMAGUCHI Yoshikazu  早稲田大学, 商学学術院, 准教授 (30534044)

Project Period (FY) 2017-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords漸近挙動 / ライデマイスタートーション / 力学系 / ゼータ関数 / 位相不変量 / 双曲多様体 / 幾何構造 / 基本群の表現 / ザイフェルト多様体 / オービフォールド / セルバーグ・ゼータ関数 / 双曲構造 / セルバーグ跡公式 / 測地線流 / 双曲曲面 / アノソフ流 / 幾何学構造 / 幾何学
Outline of Final Research Achievements

We can define a sequence of topological invariants called Reidemeister torsions for a 3-manifold and its geometric structure. I studied the asymptotic behavior for the sequence of Reidemeister torsions using a dynamical zeta function of a 3-manifold. In this study, I revealed that the dynamical zeta function defined by the geodesic flow on a 2-dimensional orbifold gives the Reidemeister torsion for unit tangent bundle over the orbifold. Moreover I have described the asymptotic behavior of higher-dimensional Reidemeister torsions for a unit tangent bundle over a 2-dimensional orbifold as the limit of the dynamical zeta function. According to the observation by the dynamical zeta function, I also presented how we can derive the area or Euler characteristic of the orbifold from the asymptotic behavior of Reidemeister torsions.

Academic Significance and Societal Importance of the Research Achievements

三次元多様体とその幾何構造が定めるライデマイスタートーションという位相不変量の系列と多様体の体積といった解析的な量の関係は双曲三次元多様体においては力学系のゼータ関数を利用することで研究されていた。本研究では非双曲三次元多様体(ザイフェルト多様体)においても、その幾何構造に関わるライデマイスタートーションの漸近挙動を力学系のゼータ関数を利用した解析的な手法で記述・考察することが可能であり、錐特異点をもつ曲面の面積が導出される現象を明らかにした。代数的・組み合わせ的な手法で主に研究されてきた非双曲三次元多様体(ザイフェルト多様体)においても解析的な手法の有効性を示した点に学術的意義がある。

Report

(7 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (17 results)

All 2022 2021 2020 2018 2017 Other

All Int'l Joint Research (2 results) Journal Article (4 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 4 results,  Acknowledgement Compliant: 1 results) Presentation (8 results) (of which Int'l Joint Research: 1 results,  Invited: 7 results) Remarks (2 results) Funded Workshop (1 results)

  • [Int'l Joint Research] The University of Texas at Dallas(米国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] The University of Texas at Dallas(米国)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Dynamical zeta functions for geodesic flowsand the higher-dimensional Reidemeister torsionfor Fuchsian groups2022

    • Author(s)
      Yamaguchi Yoshikazu
    • Journal Title

      Journal fur die reine und angewandte Mathematik (Crelles Journal)

      Volume: 784 Issue: 784 Pages: 155-176

    • DOI

      10.1515/crelle-2021-0075

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] A surgery formula for the asymptotics of the higher-dimensional Reidemeister torsion and Seifert fibered spaces2017

    • Author(s)
      Yoshikazu Yamaguchi
    • Journal Title

      Indiana University Mathematics Journal

      Volume: 2 Issue: 2 Pages: 463-493

    • DOI

      10.1512/iumj.2017.66.6012

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Higher dimensional twisted Alexander polynomials for metabelian representations2017

    • Author(s)
      Anh T. Tran, Yoshikazu Yamaguchi
    • Journal Title

      Topology and its Applications

      Volume: 229 Pages: 42-54

    • DOI

      10.1016/j.topol.2017.07.003

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The asymptotics of the higher dimensional Reidemeister torsion for exceptional surgeries along twist knots2017

    • Author(s)
      Anh T. Tran and Yoshikazu Yamaguchi
    • Journal Title

      Canadian Mathematical Bulletin

      Volume: 印刷中 Issue: 1 Pages: 211-224

    • DOI

      10.4153/cmb-2017-021-5

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups2022

    • Author(s)
      山口 祥司
    • Organizer
      トポロジー火曜セミナー
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups2021

    • Author(s)
      山口 祥司
    • Organizer
      京都大学微分トポロジーセミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups2020

    • Author(s)
      山口 祥司
    • Organizer
      東京電機大学 数学講演会
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups2020

    • Author(s)
      山口 祥司
    • Organizer
      日本数学会東北支部会(特別講演)
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Anosov flow and the higher-dimensional Reidemeister torsion2018

    • Author(s)
      山口 祥司
    • Organizer
      Workshop「New development of low-dimensional topology」
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted Alexander invariants and metabelian representations2018

    • Author(s)
      山口 祥司
    • Organizer
      Friday Seminar on Knot Theory
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] ねじれアレキサンダー不変量の漸近挙動と結び目の外部空間の幾何構造について2018

    • Author(s)
      山口 祥司
    • Organizer
      筑波大学トポロジーセミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Higher-dimensional twisted Alexander invariants for metabelian representations2017

    • Author(s)
      山口 祥司
    • Organizer
      日本数学会トポロジー分科会一般講演
    • Related Report
      2017 Research-status Report
  • [Remarks] 山口祥司のウェブサイト

    • URL

      http://www.gipc.akita-u.ac.jp/~shouji/index_ja.html

    • Related Report
      2020 Research-status Report
  • [Remarks] Yoshikazu YAMAGUCHI's web site

    • URL

      http://www.gipc.akita-u.ac.jp/~shouji/index_ja.html

    • Related Report
      2019 Research-status Report 2017 Research-status Report
  • [Funded Workshop] Volume Conjecture in Tokyo2018

    • Related Report
      2018 Research-status Report

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Published: 2017-04-28   Modified: 2024-01-30  

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