| Project/Area Number |
13640057
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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 | Tohoku University |
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Principal Investigator |
IZEKI Hiroyasu Tohoku University, Mathematical Institute, Ass. Prof., 大学院・理学系研究科, 助教授 (90244409)
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| Co-Investigator(Kenkyū-buntansha) |
FUJIWARA Koji Tbhoku University, Mathematical Institute, Ass. Prof., 大学院・理学系研究科, 助教授 (60229078)
KOTANI Motoko Tohoku University, Mathematical Institute, Ass. Prof., 大学院・理学系研究科, 助教授 (50230024)
SUNADA Toshikazu Tohoku University, Mathematical Institute, Prof., 大学院・理学系研究科, 教授 (20022741)
NAYATANI Shin Nagoya University, Graduate School of Mathematics, Ass. Prof., 大学院・多元数理科学研究科, 助教授 (70222180)
NAKAGAWA Yasuhiro Tohoku University, Mathematical Institute, Lect., 大学院・理学系研究科, 講師 (90250662)
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| Project Period (FY) |
2001 – 2002
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| Keywords | discrete groups / rigidity / index theorem / harmonic map / conformally flat / Kleinian group |
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| Research Abstract |
The purpose of this project was to investigate the rigidity of discrete groups from the viewpoint of geometry of the ideal boundary of negatively curved spaces and the cohomology of discrete groups. Our main result is summarized as follows.
Let Γ be a Kleinian group acting on n-sphere. If Γ is convex cocompact, the quotient of the domain of discontinuity is compact by definition. However, the converse is not true in general. Izeki (head investigator) showed that if the Hausdorff dimension of the limit set of Γ is less than n/2 and the quotient of the domain of discontinuity is compact, then Γ is convex cocompact. As a consequence, such a Γ is quasiconformally stable. We also gave several applications to topology and geometry of conformally flat manifolds with positive scalar curvature. In case the Hausdorff dimension of the limit set is less than (n - 2) /2, we found a proof using the index theorem for higher A-genus.
We also developed another approach to rigidity problems, which uses harmonic maps from a simplicial complex to a negatively curved metric space. We obtained a fixed-point theorem for a lattice in a p-adic Lie group, which should be regarded as a generalization of Margulis superrigidity.
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