coarse geometry of negatively curved spaces beyond relatively hyperbolic groups

• 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
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 粗幾何学 / coarse geometry / coarse Baum-Connes予想 / coarsely convex space / coares Baum-Connes / nonpositive curvature / Cartan-Hadamard theorem / 粗Cartan-Hadaramの定理 / 粗凸空間 / 粗Baum-Connes予想 / Coarse Geometry / non-positively curved / Coarse Baum-Connes予想 / 相対双曲群

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.

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．

Research Products

• [Journal Article] The coarse Baum-Connes conjecture for Busemann non-positively curved spaces (2016)
  • Author(s): Tomohiro Fukaya, Shin-ichi Oguni
  • Journal Title: Kyoto Journal of Mathematics Volume: 56(1) Issue: 1 Pages: 1-12
  • DOI: 10.1215/21562261-3445129
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Coarsely convex spaces (2019)
  • Author(s): Tomohiro Fukaya
  • Organizer: AMS Spring Central and Western Joint Sectional Meeting，Special Session on Coarse Geometry, Index Theory, and Operator Algebras: Around the Mathematics of John Roe
  • Int'l Joint Research / Invited
• [Presentation] Coarsely convex spaces (2018)
  • Author(s): Tomohiro Fukaya
  • Organizer: Rigidity School ― The Final Meeting
  • Int'l Joint Research / Invited
• [Presentation] A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture (2018)
  • Author(s): Tomohiro Fukaya
  • Organizer: Noncommutative Geometry and K-Theory at Rits -The Fourth China-Japan Conference-
  • Int'l Joint Research / Invited
• [Presentation] 粗幾何学における非正曲率空間の新しい定式化とその境界 (2018)
  • Author(s): 深谷友宏
  • Organizer: 日本数学会年会・幾何学分科会
• [Presentation] A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture (2018)
  • Author(s): Tomohiro Fukaya
  • Organizer: Noncommutative Geometry and K-Theory at Rits -The Fourth China-Japan Conference
  • Int'l Joint Research / Invited
• [Presentation] A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture (2017)
  • Author(s): Tomohiro Fukaya
  • Organizer: Workshop: Non-Positively Curved Groups and Spaces
  • Int'l Joint Research
• [Presentation] The coarse Baum-conjecture for product of nonpositive curved spaces and groups (2016)
  • Author(s): Tomohiro Fukaya
  • Organizer: Glances@Manifolds II
  • Place of Presentation: Krakow, Poland
  • Int'l Joint Research
• [Presentation] The coarse Baum-conjecture for product of nonpositive curved spaces and groups (2016)
  • Author(s): Tomohiro Fukaya
  • Organizer: The second China-Japan geometry conference
  • Place of Presentation: 中国福建省福州市
  • Int'l Joint Research / Invited
• [Presentation] The coarse Baum-conjecture for product of nonpositive curved spaces and groups (2016)
  • Author(s): Tomohiro Fukaya
  • Organizer: Workshop on Development of new methods in Symplectic Geometry: JSPS Bilateral Joint Research Project between Belgium and Japan
  • Place of Presentation: 東北大学，仙台市
  • Int'l Joint Research / Invited
• [Presentation] Boundary of relatively hyperbolic groups and coarse geoemtry (2015)
  • Author(s): Tomohiro FUKAYA
  • Organizer: 日中非可換幾何学研究集会
  • Place of Presentation: 重慶大学
  • Year and Date: 2015-12-22
  • Int'l Joint Research / Invited
• [Presentation] 相対双曲群の境界と粗幾何学 (2015)
  • Author(s): 深谷友宏
  • Organizer: 日本数学会秋季総合分科会特別講演
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-16
  • Invited