| Project/Area Number |
18K03331
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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 12010:Basic analysis-related
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| Research Institution | Shimane University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | タイヒミュラー空間 / クライン群 / リーマン面 / 双曲幾何 / 双曲多様体 / 離散群 / 双曲幾何学 / 写像類群 / 離散群論 / ３次元双曲多様体 / 複素力学系 |
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| Outline of Final Research Achievements |
The Teichmuller space of a surface of finite topological type admits a global coordinate system where parameters are lengths of some closed geodesics on the surface. We use this coordinate system with a suitable choice of closed geodesics for the Teichmuller space of the twice punctured torus and obtained a Fuchsian (matrix) representation of a twice punctured torus group and also a rational representation of its mapping class group. When a Fuchsian group of a finite coarea is given, a point of the hyperbolic plane is called exceptional if the number of sides of the Dirichlet fundamental polygon centered at this point is not maximal. With a joint work with Akira Ushijima, we show that the set of exceptional points is uncountable and generalize a result by J. Fera for cocompact Fuchsian group.
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| Academic Significance and Societal Importance of the Research Achievements |
我々の研究は，写像類のタイヒミュラー空間への作用を具体的な有理変換として表現するもので，写像類の反復合成による軌道やその不動点の近傍における挙動などを実際の数値や有理写像をともなって計算することができる。まだ大きくは進歩していないが，タイヒミュラー空間や多変数複素力学系の新しい分野を拓くものである。また3次元双曲多様体論，不連続群論や整数論などに多くの応用をもち，今後の発展が他分野に与える影響は大きいと予想する。
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