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幾何学的群論における新しい指導的理論の確立

Research Project

Project/Area Number 20H00114
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Review Section Medium-sized Section 11:Algebra, geometry, and related fields
Research InstitutionKyoto University

Principal Investigator

藤原 耕二  京都大学, 理学研究科, 教授 (60229078)

Co-Investigator(Kenkyū-buntansha) 小沢 登高  京都大学, 数理解析研究所, 教授 (60323466)
塩谷 隆  東北大学, 理学研究科, 教授 (90235507)
山下 靖  中央大学, 理工学部, 教授 (70239987)
Project Period (FY) 2020-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2024)
Budget Amount *help
¥43,160,000 (Direct Cost: ¥33,200,000、Indirect Cost: ¥9,960,000)
Fiscal Year 2024: ¥8,710,000 (Direct Cost: ¥6,700,000、Indirect Cost: ¥2,010,000)
Fiscal Year 2023: ¥8,580,000 (Direct Cost: ¥6,600,000、Indirect Cost: ¥1,980,000)
Fiscal Year 2022: ¥8,450,000 (Direct Cost: ¥6,500,000、Indirect Cost: ¥1,950,000)
Fiscal Year 2021: ¥8,580,000 (Direct Cost: ¥6,600,000、Indirect Cost: ¥1,980,000)
Fiscal Year 2020: ¥8,840,000 (Direct Cost: ¥6,800,000、Indirect Cost: ¥2,040,000)
Keywords幾何学的群論 / 作用素環論 / リーマン幾何 / 距離幾何 / 双曲群 / 作用素環 / 増大度 / 相対双曲群 / ファレル・ジョーンズ予想 / 写像類群 / アレクサンドロフ空間 / 漸近次元 / 双曲幾何
Outline of Research at the Start

幾何学的群論は1980年代にグロモフが双曲群の理論を創出したことに始まり、新たな研究分野として確立し多くの成果を生んできた。顕著な成果として、Selaによる自由群のロジックについてのTarski予想の解決、Agol-Wiseによる、3次元双曲多様体に関するバーチャルHaken予想をふくむいくつかの予想の解決などがある。
代表者は、Bestvina-Brombergとの共同研究で、擬ツリーへの群作用の構成について公理的な手法を発見した。これを基に、幾何学的群論のあらたな指導原理となるような理論と手法を確立し、重要な未解決問題を解決するのが本研究の目的である。
課題の一つとして、有限生成群の指数増大度に取り組む。

Outline of Annual Research Achievements

有限生成群の増大度は古くから研究されている重要な研究課題である。リーマン多様体の基本群の増大度は、その曲率と深い関係があり、負曲率多様体においては基本群は指数増大度を持つことがミルナーによって示されている。双曲群については、指数増大度を持つことが知られている。
有限生成群Gについて、そのすべての有限生成元集合Sに関する増大度のなす集合を、Gの増大度集合とよび、X(G)と表す。本課題では、X(G)についての基本的な理論を構築することを1つの大きな研究目標とする。これは、幾何学的群論の研究において新たな視点を提供する重要な研究であると位置づけられる。
代表者はSelaとの共同研究において、Gが非初等的な双曲群であるとき、X(G)が整列集合であることを示した。証明には、Limit groupの理論を援用した。この研究は重要な研究成果として国際的に評価されている。この研究について、いくつかの国際研究集会で講演した。
また、この研究をより広い範囲の群に拡張することを目指している。本年度は、相対双曲群や3次元多様体の基本群について、X(G)の整列性を示し、それを論文としてまとめプレプリントとして発表している。また、いくつかの国際研究集会において発表した。
ファレル・ジョーンズ予想はトポロジー・離散群における重要の未解決予想の一つである。これの重要なケースについて、Bestvinaらとの共同研究において、予想を肯定的に解決した。その論文は専門誌に出版された。

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

有限生成群の増大度全体がなす集合の整列性について、いくつかの重要なケースで顕著な結果を得たので。また、いくつかの国際研究集会で招待講演をしたので。

Strategy for Future Research Activity

有限生成群の増大度集合の整列性の研究を発展させる。特に、個別の群でなく、ある性質を満たす群の族について、増大度集合の整列性の研究を行う。

Report

(4 results)
  • 2022 Annual Research Report
  • 2021 Annual Research Report
  • 2020 Comments on the Screening Results   Annual Research Report
  • Research Products

    (38 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (6 results) Journal Article (11 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 11 results,  Open Access: 3 results) Presentation (17 results) (of which Int'l Joint Research: 13 results,  Invited: 16 results) Book (2 results) Remarks (2 results)

  • [Int'l Joint Research] ヘブライ大(イスラエル)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] ユタ大(米国)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] ケンブリッジ大(英国)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] ユタ大(米国)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] レンヌ大学(フランス)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] ヘブライ大学(イスラエル)

    • Related Report
      2020 Annual Research Report
  • [Journal Article] The Farrell-Jones conjecture for hyperbolic-by-cyclic groups2023

    • Author(s)
      Bestvina, Mladen; Fujiwara, Koji; Wigglesworth, Derrick
    • Journal Title

      Int. Math. Res. Not. IMRN

      Volume: 7 Issue: 7 Pages: 5887-5904

    • DOI

      10.1093/imrn/rnac012

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The rates of growth in a hyperbolic group2023

    • Author(s)
      Fujiwara, Koji; Sela, Zlil
    • Journal Title

      Invent. Math

      Volume: 233 Issue: 3 Pages: 1427-1470

    • DOI

      10.1007/s00222-023-01200-w

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A natural compactification of the Gromov--Hausdorff space2023

    • Author(s)
      Nakajima Hiroki、Shioya Takashi
    • Journal Title

      Geometriae Dedicata

      Volume: 218 Issue: 1 Pages: 18-18

    • DOI

      10.1007/s10711-023-00852-5

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the joint spectral radius for isometries of non-positively curved spaces and uniform growth.2021

    • Author(s)
      Breuillard, Emmanuel; Fujiwara, Koji
    • Journal Title

      Ann. Inst. Fourier

      Volume: 71 Issue: 1 Pages: 317-391

    • DOI

      10.5802/aif.3374

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Proper actions on finite products of quasi-trees.2021

    • Author(s)
      Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji
    • Journal Title

      Ann. H. Lebesgue

      Volume: 4 Pages: 685-709

    • DOI

      10.5802/ahl.85

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On characterizations of amenable C*-dynamical systems and new examples2021

    • Author(s)
      N. Ozawa and Y. Suzuki
    • Journal Title

      Selecta Mathematica

      Volume: 27 Issue: 5 Pages: 1-29

    • DOI

      10.1007/s00029-021-00699-2

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Graph manifolds as ends of negatively curved Riemannian manifolds2020

    • Author(s)
      Fujiwara Koji、Shioya Takashi
    • Journal Title

      Geometry & Topology

      Volume: 24 Issue: 4 Pages: 2035-2074

    • DOI

      10.2140/gt.2020.24.2035

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Solvable groups of interval exchange transformations.2020

    • Author(s)
      Dahmani, Francois; Fujiwara, Koji; Guirardel, Vincent
    • Journal Title

      Ann. Fac. Sci. Toulouse Math.

      Volume: 29 Issue: 3 Pages: 595-618

    • DOI

      10.5802/afst.1641

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras2020

    • Author(s)
      M. Mori and N. Ozawa
    • Journal Title

      Studia Mathematica

      Volume: 250 Issue: 3 Pages: 265-281

    • DOI

      10.4064/sm180727-14-11

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Full factors and co-amenable inclusions2020

    • Author(s)
      J. Bannon, A. Marrakchi, and N. Ozawa
    • Journal Title

      Communications in Mathematical Physics

      Volume: 378 Issue: 2 Pages: 1107-1121

    • DOI

      10.1007/s00220-020-03816-y

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] An entropic proof of cutoff on Ramanujan graphs2020

    • Author(s)
      N. Ozawa
    • Journal Title

      Electron. Commun. Probab

      Volume: 77 Issue: none

    • DOI

      10.1214/20-ecp358

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Presentation] Growth rates in a family of hyperbolic groups.2024

    • Author(s)
      Koji Fujiwara
    • Organizer
      International Colloquium on randomness, geometry, and dynamics. IISER, Pune.India.
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth rates in hyperbolic groups.2023

    • Author(s)
      Koji Fujiwara
    • Organizer
      Groups and Dynamics in Geometry. Monte Verita, Switzerlad.
    • Related Report
      2022 Annual Research Report
  • [Presentation] Growth rates in hyperbolic groups. Groups and groups in the South East. Oxford. UK.2023

    • Author(s)
      Koji Fujiwara
    • Organizer
      Groups and groups in the South East. Oxford. UK.
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth rates in hyperbolic groups.2023

    • Author(s)
      Koji Fujiwara
    • Organizer
      Frontiers of Riemannian Geometry, Shioya 60. 東北大
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The asymptotic dimension of arc graphs.2022

    • Author(s)
      Koji Fujiwara
    • Organizer
      "Mapping class groups and Out(Fn)" IHP, France.
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth of acylindrically hyperbolic groups.2022

    • Author(s)
      Koji Fujiwara
    • Organizer
      "Hyperbolic groups and their generalizations", IHP, France.
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 2022.9.2 双曲群の増大度のなす集合2022

    • Author(s)
      Koji Fujiwara
    • Organizer
      幾何学シンポジウム。東京理科大
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] The set of growth of a hyperbolic group.2022

    • Author(s)
      Koji Fujiwara
    • Organizer
      Symposium of the Global Math Network. Bonn,Germany.
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth of hyperbolic groups.2021

    • Author(s)
      Koji Fujiwara
    • Organizer
      Seminar "Action". ENS Lyon.
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth of hyperbolic groups.2021

    • Author(s)
      Koji Fujiwara
    • Organizer
      Geometry seminar. U Chicago
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Growth of hyperbolic groups.2021

    • Author(s)
      Koji Fujiwara
    • Organizer
      Geometry and Analysis on Groups Seminar, U of Vienna.
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Asymptotic dimension of planes.2021

    • Author(s)
      Koji Fujiwara
    • Organizer
      Differentialgeometrie im Grossen, MFO, Germany.
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Rates of growth in a hyperbolic group. Artin Groups, CAT(0) geometry and related topics,2021

    • Author(s)
      Koji Fujiwara
    • Organizer
      A conference in honor of RUTH CHARNEY. Ohio State University.
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth of hyperbolic groups.2021

    • Author(s)
      Koji Fujiwara
    • Organizer
      SFB-Lecture, Regensburg, Germany
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth of hyperbolic groups.2021

    • Author(s)
      Koji Fujiwara
    • Organizer
      Group theory seminar, ENS, Paris.
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth of hyperbolic groups.2020

    • Author(s)
      Koji Fujiwara
    • Organizer
      'Geometric Structures in Group Theory'
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth of hyperbolic groups.2020

    • Author(s)
      Koji Fujiwara
    • Organizer
      GGT in East Asia.
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Book] 幾何学入門事典2023

    • Author(s)
      藤原耕二(分担執筆)
    • Total Pages
      8
    • Publisher
      朝倉書店
    • Related Report
      2022 Annual Research Report
  • [Book] 離散群の幾何学2021

    • Author(s)
      藤原 耕二
    • Total Pages
      224
    • Publisher
      朝倉書店
    • ISBN
      9784254117554
    • Related Report
      2020 Annual Research Report
  • [Remarks] 「幾何学的群論における新しい指導的理論の確立」

    • URL

      https://www.math.kyoto-u.ac.jp/~kfujiwara/kakenA2020.html

    • Related Report
      2022 Annual Research Report
  • [Remarks] Koji Fujiwara

    • URL

      https://www.math.kyoto-u.ac.jp/~kfujiwara/

    • Related Report
      2021 Annual Research Report

URL: 

Published: 2020-04-28   Modified: 2024-12-25  

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