2023 Fiscal Year Final Research Report
New development of the coarse geometry of nonpositively curved spaces
| Project/Area Number |
19K03471
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Tokyo Metropolitan University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 粗幾何学 / 粗凸空間 / 粗Baum-Connes予想 / 非正曲率空間 |
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| Outline of Final Research Achievements |
Gromov hyperbolic spaces are analogous to simply connected Riemannian manifolds of negative curvature, and there are many studies from the viewpoint of geometric group theory and noncommutative geometry. Recently, studies of "spaces of nonpositive curvatures" become very active. With Shin-ichi Oguni, we introduced "coarsely convex spaces", which include many spaces of nonpositive curvatures, like Gromov hyperbolic spaces, CAT(0) spaces, systolic complexes, and proper injective metric spaces. In 2017, We proved the coarse Baum-Connes conjecture for proper coarsely convex spaces with minimal knowledge of these spaces. In this project, we reconstructed the basic theory of coarsely convex spaces.
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| Free Research Field |
幾何学的群論
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| Academic Significance and Societal Importance of the Research Achievements |
単連結完備負曲率リーマン多様体の粗幾何学における類似物が、Gromov双曲空間であり、これまで幾何学的群論や非可換幾何学の観点から数多の研究が為されて きた。近年、「負曲率」を「非正曲率」に置き換えた、様々な距離空間のクラスが活発に研究されている。尾國新一氏との共同研究で2017年に導入した粗凸空間、はそうした空間の多くを包含する非正曲率空間のクラスである。これまで、Gromov双曲空間、CAT(0)空間、systolic 複体、injective metric spacesなど、個別の設定で行われてきた「非正曲率距離空間」の研究を、粗凸空間という設定の元で、統一的な理論の構築を進められた。
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