2021 Fiscal Year Final Research Report

Progress in Geometric group theory

Research Project

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Project/Area Number	15H05739
Research Category	Grant-in-Aid for Scientific Research (S)
Allocation Type	Single-year Grants
Research Field	Geometry
Research Institution	Kyoto University
Principal Investigator	Fujiwara Koji 京都大学, 理学研究科, 教授 (60229078)
Co-Investigator(Kenkyū-buntansha)	山口孝男京都大学, 理学研究科, 教授 (00182444) 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)
Project Period (FY)	2015-05-29 – 2020-03-31
Keywords	写像類群 / 擬ツリー / カズダンの性質T / 射影複体 / 自由群の自己同型群
Outline of Final Research Achievements	During this project, Fujiwara with his joint work with Bestvina-Bromberg introduced the theory of Projection complex, and found several important applications. For example, we proved that a mapping class group acts on a finite product of quasi-tree with a QI-embedding orbit, and as a consequence it has finite asymptotic dimension. By now the technique of projection complex became an important tool in Geometric group theory. Ozawa in his joint work with Kaluba-Novak proved that the automorphism group of the free group of rank 5 has property T, using computer. This settles a long standing problem. It also opens a new direction of research.
Free Research Field	幾何学的群論
Academic Significance and Societal Importance of the Research Achievements	これらの結果は幾何学的群論の画期的な成果と評価されている。たとえば、代表者は２０１８年に開催された国際数学者会議で、射影複体の理論とその写像類群への応用について招待講演をした。また、代表者は２０１５年度日本数学会秋季賞を受賞した。小澤が証明した５次自由群の自己同型群が性質Tをもつ、という定理は、長年の未解決問題に決着をつけるだけでなく、その手法において斬新で新しい研究の方向性を開いた点でも画期的である。