2003 Fiscal Year Final Research Report Summary
Asymptotic structures of non-compact hyperbolic 3-manifoIds and differential geometry
| Project/Area Number |
12640063
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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 |
Geometry
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| Research Institution | Osaka University (2001-2003)
The University of Tokyo (2000) |
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Principal Investigator |
OHSHIKA Ken'ichi Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70183225)
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| Co-Investigator(Kenkyū-buntansha) |
HARA Yasuhiro Osaka University Graduate School of Science, Instructor, 大学院・理学研究科, 助手 (10294141)
NAGASAKI Ikumitsu Osaka University Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (50198305)
ENDO Hisaaki Osaka University Graduate School of Science, Ass.Professor, 大学院・理学研究科, 助教授 (20323777)
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| Project Period (FY) |
2000 – 2003
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| Keywords | Hyperbolic 3-manifold / Kleinian group / Deformation space / ending lamination / tameness |
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| Research Abstract |
We have been studying the topological properties of the deformation spaces of hyperbolic structures on 3-manifolds making use of differential geometry and low-dimensional manifold theory. In particular, we investigated the behavior of hyperbolic structures on the boundaries of quasi-conformal deformation spaces, which is known to coincide with the boundaries of the entire deformation spaces. As a result of this line of research, Ohshika proved that Bers-Thurston conjecture, which states that every finitely generated Kleinian group would be an algebraic limit of quasi-conformal deformations of a geometrically finite group, follows once Marden's tameness conjecture is proved to be true. This should be an important progress towards the solution of the ultimate problem of classifying the hyper-bolic structures on a given 3-manifold with finitely generated fundamental group.
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