2006 Fiscal Year Final Research Report Summary
Research on Jorgensen groups and classical Schottky groups
| Project/Area Number |
16540147
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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, Faculty of Science, Professor, 理学部, 教授 (40022222)
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| Co-Investigator(Kenkyū-buntansha) |
NAKANISHI Toshihiro Shimane University, Interdisciplinary Faculty of Science and Engineering, Professor, 総合理工学部, 教授 (00172354)
OKUMURA Yoshihide Shizuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (90214080)
KOYAMA Akira Shizuoka University, Faculty of Science, Professor, 理学部, 教授 (40116158)
KUMURA Hironori Shizuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (30283336)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Jorgensen group / Jorgensen number / classical Schottky group / Schottky group / Kleinian group / Schottky space / Jorgensen's inequality / uniformization of a Riemann surface |
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| Research Abstract |
We have studied the following four themes from 2004 to 2006. 1. Jorgensen groups. 2. The Whitehead link group. 3. Jorgensen numbers. 4. Classical Schottky groups.
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 found all Jorgensen groups during 2004, and 2006. The results were talked at the Mathematical Society of Japan, the RIMS, Kyoto university and the International congress of Mathematicians in Joensuu, Finland. Furthermore, they were published from Osaka J. Math. Kodai Math. J. and Comput Methods Funct. Theory.
2. 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. This result was published at Boletin de Soc. Mat. Mex.
3. Jorgensen numbers. We showed that for every natural number and every real number greater than four there exists a classical Schottky group whose Jorgensen number is the number.
4. Classical Schottky groups. We did not obtain any meaningful results.
We are planning the following : (1) To study structures of 3-manifolds represented by Jorgensen groups (2) To find all Jorgensen groups of elliptic type (3) To find Jorgensen numbers of (classical) Schottky groups and (4) To study a uniformization of a Riemann surfaces by classical Schottky groups.
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