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On continuity of growth rates and spectral radii on the space of marked groups

Research Project

Project/Area Number 20K14318
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11020:Geometry-related
Research InstitutionAshikaga University (2023)
Waseda University (2020-2022)

Principal Investigator

Yukita Tomoshige  足利大学, 工学部, 講師 (80843903)

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2023: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywordsコクセター系 / 増大度 / Salem数 / Pisot数 / Perron数 / 双曲幾何 / 双曲多面体 / コクセター群 / 標識付き群 / 幾何学的群論 / スペクトル半径 / 双曲群 / 双曲幾何学
Outline of Research at the Start

有限生成群Gと順序を付けた有限生成系Sの組(G, S)を標識付き群という。標識付き群の全体にはCayleyグラフを用いた距離が定まる。標識付き群(G, S)に対して定まる増大度ω(G, S)とスペクトル半径ρ(G, S)の標識付き群の空間M上での連続性を明らかにすることが本研究の目的である。特に、コクセター群の全体C上での増大度およびスペクトル半径の振る舞いを明らかにすることを研究目標とする。

Outline of Final Research Achievements

We considered the set of Coxeter systems with N generators as a subspace of the space of marked groups. Then we showed that the space of Coxeter systems is compact and the growth rates are continuous as a function on the space. As an application of this result, we studied arithmetic nature of the growth rates of Coxeter systems. We showed that if the Euler characteristic of the nerve of a 2-dimensional Coxeter system is positive (resp. zero), then the growth rate is a Salem number (resp. Pisot number). These results are extensions of the previous results on the growth rates of discrete hyperbolic reflection groups.

Academic Significance and Societal Importance of the Research Achievements

有限生成群の増大度関数や増大度についての研究は、MilnorのRiemann多様体と基本群の関係についての研究から始まり、Gromovらにより行われた。特に、多項式増大を持つ群は有限指数のベキ零部分群をもつというGromovの多項式増大定理は好例である。本研究では、コクセター系に焦点を当てて増大度についての研究を行い、特に標識付き群を変数とする関数としての連続性や数論的性質について、双曲幾何で得られてきた結果をコクセター系に一般化したものである。

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (20 results)

All 2023 2022 2021 2020

All Journal Article (2 results) (of which Peer Reviewed: 1 results) Presentation (18 results) (of which Int'l Joint Research: 4 results,  Invited: 12 results)

  • [Journal Article] On the continuity of the growth rate on the space of Coxeter systems2023

    • Author(s)
      Tomoshige Yukita
    • Journal Title

      Groups, Geometry, and Dynamics

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Coxeter systems with 2-dimensional Davis complexes, growth rates and Perron numbers2023

    • Author(s)
      Naomi Bredon and Tomoshige Yukita
    • Journal Title

      Algebraic & Geometric Topology

      Volume: -

    • Related Report
      2022 Research-status Report
  • [Presentation] コクセター群のnerveのトポロジーと増大度について2023

    • Author(s)
      雪田友成
    • Organizer
      福岡大学微分幾何セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] コクセター系の増大度とナーブの位相型について2023

    • Author(s)
      雪田友成
    • Organizer
      金沢トポロジーセミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] コクセター系の極限と増大型について2023

    • Author(s)
      雪田友成
    • Organizer
      Random Topics on Teichmuller Theory II
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] 双曲多面体の変形とコクセター系の空間について2022

    • Author(s)
      雪田友成
    • Organizer
      N-KOOK seminar
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Arithmetic nature and continuity of growth rates of Coxeter systems2022

    • Author(s)
      雪田友成
    • Organizer
      Prossimi seminari del Dipartimento di matematica
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] コクセター系のnerveの組み合わせと増大度の数論的性質について2022

    • Author(s)
      雪田友成
    • Organizer
      早稲田双曲幾何幾何学的群論セミナー
    • Related Report
      2022 Research-status Report
  • [Presentation] 2次元コクセター系のオイラー数と増大度の数論的性質2022

    • Author(s)
      雪田友成
    • Organizer
      東工大複素解析セミナー
    • Related Report
      2022 Research-status Report
  • [Presentation] コクセター系の指数増大度とSalem数およびPisot数について2022

    • Author(s)
      雪田友成
    • Organizer
      第6回幾何学的群論ワークショップ
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] コクセター系のnerveのオイラー標数と増大度2022

    • Author(s)
      雪田友成
    • Organizer
      2022年度「リーマン面・不連続群論」研究集会
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Convergent sequences of Coxeter groups and its growth rates2021

    • Author(s)
      雪田友成
    • Organizer
      2020年度「リーマン面・不連続群論」研究集会
    • Related Report
      2021 Research-status Report 2020 Research-status Report
    • Invited
  • [Presentation] 局所剛性を持つ5次元双曲離散鏡映群2021

    • Author(s)
      雪田友成
    • Organizer
      日本数学会 ・2021年度年会
    • Related Report
      2021 Research-status Report
  • [Presentation] コクセター群の増大度の連続性2021

    • Author(s)
      雪田友成
    • Organizer
      日本数学会 ・2021年度年会
    • Related Report
      2021 Research-status Report
  • [Presentation] Local rigidity of right-angled Coxeter groups in hyperbolic 5-space2021

    • Author(s)
      雪田友成
    • Organizer
      RIMS Workshop Geometry of discrete groups and hyperbolic spaces
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth rates and spectral radii of Coxeter groups2021

    • Author(s)
      雪田友成
    • Organizer
      World of Group Craft
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Topology of the space of Coxeter systems and growth rates2021

    • Author(s)
      雪田友成
    • Organizer
      奈良双曲幾何セミナー
    • Related Report
      2021 Research-status Report
  • [Presentation] Continuity of the growth rate on the space of Coxeter groups2020

    • Author(s)
      雪田友成
    • Organizer
      Geometric Group Theory in East Asia
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Rigidity of hyperbolic reflection groups in dimension 4 and 52020

    • Author(s)
      雪田友成
    • Organizer
      第63回函数論シンポジウム
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] コクセター群の空間と増大度の連続性2020

    • Author(s)
      雪田友成
    • Organizer
      早稲田双曲幾何幾何学的群論セミナー
    • Related Report
      2020 Research-status Report

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Published: 2020-04-28   Modified: 2025-01-30  

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