2023 Fiscal Year Final Research Report
Study of geometric structures of 3-dimensional cone hyperbolic manifolds using the families of canonical fundamental polyhedra
| Project/Area Number |
19K03497
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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 | Osaka Metropolitan University (2022-2023)
Osaka City University (2019-2021) |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 双曲幾何 / 錐多様体 / 基本多面体 / 実射影構造 |
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| Outline of Final Research Achievements |
The main results obtained in this project can be roughly divided into three categories: a detailed proof of a definitive property for non-free two-parabolic Klein groups, which includes the Riley slice; observations and related numerical experiments on the stability of the combinatorial structure of canonical fundamental polyhedra for a certain families of cone hyperbolic 3-manifolds; and introducing a new viewpoint on the analysis of the combinatorial structure of the ends, which is a bottleneck in the study of noncompact hyperbolic manifolds from the viewpoint of fundamental polyhedra. The new viewpoint is closely related to the study of real projective manifolds, and is expected to form an important basis for future research on degenerations and transitions of infinite volume hyperbolic structures from the viewpoint of real projective geometry.
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| Free Research Field |
低次元多様体
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| Academic Significance and Societal Importance of the Research Achievements |
本研究では3次元双曲幾何を中心的な研究対象とする．3次元空間内の結び目はDNAや高分子などの数学モデルのうち最も基本的なものと考えられるが，その位相的構造の解明はそれら具体的な実在を対象とする自然科学分野においても重要な役割を担っている．非常に多くの結び目が双曲幾何により支配され，その幾何構造は標準的な基本多面体により組合せ的に理解可能であることが知られており，本研究はその理解をさらに進めるための一ステップである．本研究で発見された新たな視点は，そうした組合わせ構造がより広い数学的対象に対しても有効な手掛かりとなることを示唆するという意味で大変興味深いものである．
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