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

2021 Fiscal Year Final Research Report

• 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
• 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

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.

Free Research Field

幾何学

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。