2003 Fiscal Year Final Research Report Summary
Research on Jorgensen groups and Schottky spaces
| Project/Area Number |
14540170
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Shizuoka University |
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Principal Investigator |
SATO Hiroki Shizuoka University, Science, Professor, 理学部, 教授 (40022222)
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| Co-Investigator(Kenkyū-buntansha) |
AKUTAGAWA Kazuo Shizuoka University, Science, Associate Professor, 理学部, 助教授 (80192920)
OKUMURA Yoshihide Shizuoka University, Science, Associate Professor, 理学部, 助教授 (90214080)
NAKANISHI Toshihiro Nagoya University, Mathematics, Associate Professor, 大学院・多元数理科学研究所, 助教授 (00172354)
OKUYAMA Yuusuke kanazawa University, Science, Lecturer, 理学部, 講師 (00334954)
KUMURA Hironori Shizuoka University, Science, Associate Professor, 理学部, 助教授 (30283336)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Jorgensen group / Jorgensen number / Jorgensen's inequality / Whitehead link group / Schottky space / Schottky group / Kleinian group / Uniformization of Riemann surface |
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| Research Abstract |
We have studied the following four themes from 2002 to 2003. 1.Jorgensen groups. 2.The Picard group. 3.The Whitehead link group. 4.Classical Schottky spaces and Jorgensen number.
1.Jorgensen groups. A Jorgensen group is a non-elementary two-generator discrete group whose Jorgensen number is one. There are two types -parabolic type and elliptic type-for Jorgensen groups. Here we considered of parabolic type. There are three types for Jorgensen groups of parabolic type (finite type, countably infinite type and uncountably infinite type). We obtained the following. (1)We found all Jorgensen groups of finite type and all Jorgensen groups of countably infinite type in 2002, and (2)we found all Jorgensen groups of uncountably infinite type in 2003. Consequently we found all Jorgensen groups of parabolic type. The results (1) was talked at the International congress of Mathematicians in Beijing in 2002, and the result (2) was talked at Peking University in 2003.
2.The Picard group. We constructed a new fundamental region for the Picard group and we found eight relations for two generators of the group by using the fundamental region. This result was published in the Proceedings of the ISAAC Congress in Berlin in 2003.
3.The Whitehead link group. We proved that the Jorgensen number of the Whitehead link is two. Therefore the Whitehead link is not a Jorgensen group. We talked this result at the Internatonal Conference of Topology in 2002.
4.Classical Schottky spaces and Jorgensen number. We showed that there exists a classical Schottky group whose Jorgensen number is a given real number j【greater than or equal】4. We will talk this result at the International Conference of Potential Theory this summer.
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Research Products
(12 results)
