Fixed point properties on Busemann Non-positively curved spaces, expanders, and generalizations

• Project/Area Number: 17H04822
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Masato Mimura 東北大学, 理学研究科, 准教授 (10641962)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥6,500,000 (Direct Cost: ¥5,000,000、Indirect Cost: ¥1,500,000); Fiscal Year 2020: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 固定点性質 / エキスパンダー族 / 剰余的有限群 / マーク付き群のなす空間 / 副有限群 / エクスパンダー / 剛性 / 極値組合せ論 / ラムゼー型現象 / Green-Taoの定理 / 離散群の剛性 / エクスパンダーグラフ / コンパクト群 / 幾何学 / 離散群

Outline of Final Research Achievements

Study on marked finite groups and fixed point properties for the limit groups have been proceeded. Upgrading theorems form relative fixed point properties to the full property have been obtained. As an application, fixed point properties for elementary groups defined over integral group rings of finitely generated groups have been proved. A compact group that admits for every countable residually finite group a finitely generated dense subgroup containing an isomorphic copy of the give group has been constructed. This dense subgroup can be taken to be a group quotient of an elementary group over some group ring. Combination of these results shows that this dense subgroup can have fixed point properties on several metric spaces.

Academic Significance and Societal Importance of the Research Achievements

Lubotzky と Weiss によって1990年代に、「与えられた無限コンパクト群の有限生成稠密部分群の離散群としての性質は、稠密部分群の取り方によってどれくらい変わりうるのか」という方向の問題が提起された。Ershov と Jaikin-Zapirain は1つの群が従順でもう一つの群が Kazhdan の性質 (T) をもつような例が構成した。本研究では群環上の基本行列群の研究を推し進めることで、Lubotzky と Weiss の問題に関して稠密部分群の群性質がさらに劇的に変わりうることを証明した。特に、(T) をもつ方の群は与えられた任意の可算な剰余的有限群を含むようにできる。

Research Products

• [Journal Article] Group approximation in Cayley topology and coarse geometry Part I: Coarse embeddings of amenable groups (2021)
  • Author(s): Masato Mimura and Hiroki Sako
  • Journal Title: Journal of Topology and Analysis Volume: 13 Issue: 01 Pages: 1-47
  • DOI: 10.1142/s1793525320500089
  • Peer Reviewed / Open Access
• [Journal Article] Amenability versus non-exactness of dense subgroups of a compact group (2019)
  • Author(s): Masato Mimura
  • Journal Title: Journal of the London Mathematical Society Volume: 100 Issue: 2 Pages: 592-622
  • DOI: 10.1112/jlms.12229
  • Peer Reviewed / Open Access
• [Journal Article] Group approximation in Cayley topology and coarse geometry, Part II: Fibred coarse embeddings (2019)
  • Author(s): Masato Mimura and Hiroki Sako
  • Journal Title: Analysis and Geometry in Metric Spaces Volume: 7 Issue: 1 Pages: 62-108
  • DOI: 10.1515/agms-2019-0005
  • Peer Reviewed / Open Access
• [Journal Article] Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles (2018)
  • Author(s): M,Mimura
  • Journal Title: Journal of the European Mathematical Society Volume: 20 Issue: 1 Pages: 103-117
  • DOI: 10.4171/jems/760
  • Peer Reviewed
• [Presentation] Avoiding a configuration, and the slice rank method for a system of equations (2020)
  • Author(s): 見村万佐人
  • Organizer: JCCA2020-DMIA2020-SGT9
  • Invited
• [Presentation] An extreme counterexample to the Lubotzky--Weiss conjecture (2019)
  • Author(s): Masato Mimura
  • Organizer: Rigidity
  • Int'l Joint Research / Invited
• [Presentation] Upgrading relative fixed points (2019)
  • Author(s): Masato Mimura
  • Organizer: Beyond Spectral Gaps
  • Int'l Joint Research / Invited
• [Presentation] 離散群からエクスパンダーグラフを作る (1) (2019)
  • Author(s): 見村万佐人
  • Organizer: Summer School 数理物理 2019
  • Invited
• [Presentation] 離散群からエクスパンダーグラフを作る (2) (2019)
  • Author(s): 見村万佐人
  • Organizer: Summer School 数理物理 2019
  • Invited
• [Presentation] 離散群からエクスパンダーグラフを作る (3) (2019)
  • Author(s): 見村万佐人
  • Organizer: Summer School 数理物理 2019
  • Invited
• [Presentation] 有限生成群のなす位相空間 (2019)
  • Author(s): 見村万佐人
  • Organizer: 一般位相幾何学の発展と諸分野との連携
  • Invited
• [Presentation] ルボツキーとヴァイスの予想とその周辺 (2019)
  • Author(s): 見村万佐人
  • Organizer: 淡路島幾何学研究集会2019
• [Presentation] Profinite actions on a common set (2018)
  • Author(s): Masato Mimura
  • Organizer: Rigidity School final
  • Int'l Joint Research / Invited
• [Presentation] Upgrading fixed points (2018)
  • Author(s): Masato Mimura
  • Organizer: Superexpanders and Their Coarse Geometry
  • Int'l Joint Research / Invited
• [Presentation] Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles (2017)
  • Author(s): M.Mimura
  • Organizer: Besancon-Neuchatel functional analysis meeting
  • Int'l Joint Research / Invited
• [Presentation] BG, or not BG, that is the question (2017)
  • Author(s): M.Mimura
  • Organizer: EGGS (Ergodic and Geometric Group Theory in Sendai)
  • Int'l Joint Research