Deformation of hyperbolic structures and geometry of non-discrete representations

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14530
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Hiroshima University (2022-2023); Saitama University (2019-2021)
• Principal Investigator: Yoshida Ken'ichi 広島大学, 持続可能性に寄与するキラルノット超物質国際研究所, 特任准教授 (70793371)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 双曲的デーン手術 / 有限被覆 / 双曲錐構造 / 指標多様体 / トーラス上の絡み目

Outline of Final Research Achievements

In this research, we geometrically investigated the representation spaces of the fundamental groups of 3-manifolds, based on deformation of incomplete hyperbolic structures, such as cone structures. As results, we obtained an example that degeneration of hyperbolic cone structures with decreasing cone angles. In connection with this, we define holed cone structure as a generalization of cone structure to avoid the intersection of singular loci during deformation.
Furthermore, we investigated doubly periodic tangles obtained as the universal coverings of links in the thickened torus. We obtained results on relations between finite coverings and isotopies of links in the thickened torus by using geometric structures on the complements of links.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

錐角減少変形での双曲錐構造の退化は特殊な現象ではないはずだが、具体的な初めての記述である。この例は双曲錐構造の変形を考える上で重要だと考えられる。錐構造の一般化は、基本群の表現を幾何学的に表すことができる範囲が増えるので、双曲錐構造の大域的な剛性を考察する上で役立つと考えられる。
また、二重周期絡み目はテキスタイルの構造を表すので、応用研究としての価値もあると考えられる。