| Project/Area Number |
11640056
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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・理学研究科, 助教授 (90244409)
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| Co-Investigator(Kenkyū-buntansha) |
AKUTAGAWA Kazuo Shizuoka University, Department of Mathematics, Ass.Prof., 理学部, 助教授 (80192920)
NAKAGAWA Yasuhiro Tohoku University, Mathematical Institute, Lect., 大学院・理学研究科, 講師 (90250662)
SUNADA Toshikazu Tohoku University, Mathematical Institute, Prof., 大学院・理学研究科, 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)
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| Project Period (FY) |
1999 – 2000
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| Keywords | discrete groups / rigidity / ideal boundary / Kleinian groups / conformally flat / index theorem / scalar curvature |
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| Research Abstract |
The purpose of this project was to investigate the stability/rigidity of discrete groups from the viewpoint of geometry of the ideal boundary of negatively curved spaces and the cohomology of deiscrete 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 applied the index theorem to the quotient of the domain of discontinuity. We note here what we mean by the ideal bounary is just the quotient of the domain of discontinuity. The higher A^^<^>-genus carries the information of the fundamental group, which turns out to be isomorphic to Γ in our case, and that is all that the higher A^^<^>-genus knows. And it is determined by the cohomology of Γ.
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