coarse geometry of negatively curved spaces beyond relatively hyperbolic groups

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17528
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tokyo Metropolitan University (2016-2018); Tohoku University (2015)
• Principal Investigator: Fukaya Tomohiro 首都大学東京, 理学研究科, 准教授 (40583456)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 粗幾何学 / coarse geometry / coarse Baum-Connes予想

Outline of Final Research Achievements

We introduced a new class of metric spaces, called "coarsely convex spaces", which is a generalization of Riemannian manifolds of nonpositive sectional curvatures. This class is closed under both quasi-isometry and direct products. We constructed boundaries of the coarsely convex spaces and we established a coarse version of the Cartan-Hadamard theorem. Using this, we proved that the coarsely convex spaces satisfy the coarse Baum-Connes conjecture.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

粗Baum-Connes予想は非可換幾何学及び距離空間の粗幾何学に於ける中心的な問題である．私達は粗凸空間という，非正曲率を持つ距離空間の新しい定式化を導入し，その空間に対して粗Baum-Connes予想が成り立つことを示した．近年，粗凸空間に幾何学的な作用を持つ様々な群が発見されている．私達の結果から，こうした群に対して粗Baum-Connes予想が成立することが従う．