On the rigidity of finitely generated groups of homomorphisms of the circle

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K23406
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: University of the Ryukyus (2022); Ehime University (2019-2021)
• Principal Investigator: Kato Motoko 琉球大学, 教育学部, 准教授 (00847593)
• Project Period (FY): 2019-08-30 – 2023-03-31
• Keywords: 固定点性質 / CAT(0)空間への群作用 / 円の自己同相写像 / Richard Thompsonの群

Outline of Final Research Achievements

On groups of homomorphisms of the circle, we studied fixed point properties of group actions on non-positively curved spaces. In this research, we showed relative fixed point properties for groups called ring groups, with respect to finitely generated subgroups of their commutator subgroups. As an application, we showed that Higman-Thompson groups T_n, which are generalizations of Richard Thompson's group T, admit fixed point properties for semi-simple actions on CAT(0) spaces of finite covering dimension. In the proof, we constructed new finite generating sets for every T_n and showed that every T_n has a structure of a ring group.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

一般に固定点性質を持つ群の具体例を構成するのは難しいが, Richard Thompson群T, Vはそのような数少ない具体例の一つとして知られている. しかし, T_nが同様の固定点性質を持つかどうかは知られていなかった.本研究では, T_nがring群の構造を持つことを示した. この過程で, T_nの新たな有限生成系を構成した. この生成系は, T_nの自己相似性を反映するという意味で性質の良いものである. さらにそれを用いて, Tに対する証明の一般化の仮定における技術的な困難を回避することに成功した.