2003 Fiscal Year Final Research Report Summary
Diffeomorphism groups --from a view point of rigidity problem
| Project/Area Number |
12440016
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Nagoya University |
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Principal Investigator |
KANAI Masahiko Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70183035)
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| Co-Investigator(Kenkyū-buntansha) |
TSUBOI Takashi The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理学研究科, 教授 (40114566)
KOTANI Motoko Tohoku University, Mathematical Institute, Associate Professor, 大学院・理学研究科, 教授 (50230024)
IZEKI Hiroyasu Tohoku University, Mathematical Institute, Associate Professor, 大学院・理学研究科, 助教授 (90244409)
FUJIWARA Koji Tohoku University, Mathematical Institute, Associate Professor, 大学院・理学研究科, 助教授 (60229078)
NAYATANI Shin Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 助教授 (70222180)
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| Project Period (FY) |
2000 – 2003
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| Keywords | Anosov systems / bounded cohomology / harmonic map / Kleinian group / large deviation / projectively Anosov flow / rigidity problem / spectrum |
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| Research Abstract |
<Kanai> He made computation of Gelfand-Fuks cohomology of diffeomorphism groups and their homogeneous spaces especially keeping in his mind possible applications of it to rigidity problems. A new perspective on infinitesimal rigidity of Anosov actions of higher-rank abelian groups that arise from semisimple Lie groups of rank greater than one has also been obtained by him.
<Izeki> It is known that the domain of discontinuity of a convex-cocompact Kleinian group is compact. Conversely, he proved that a Kleinian group is convex cocompact provided the Hausdorff dimension of the limit set is less than n/2.
<Izeki and Nayatani> They introduced a combinatorial notion of harmonic map of a simplicial complex into singular space of nonpositive curvature, and showed an existence theorem under an appropriate assumption. A fixed point theorem for an isometric action of a discrete group on a nonpositively curved space has been established as an application.
<Kotani> She investigated, in a joint work of T.Sunada, the large deviation principle. Another achievement by her is the Lipschitz continuity of the bounds of the spectrum of some self-adjoint operator with magnetic, effect.
<Tsuboi> A projective Anosov flow is said to be regular if its stable and unstable plane fields are integrated by smooth foliations. He proved that for a regular projective Anosov flow on a Seifert fibered space if the associated foliations have no compact leaf then it is a regular Anosov flow, and in consequence is quasi-Fuksian, due to a theorem of Ghys.
<Fujiwara> In the joint work with Bestvina, he made a computation of the second bounded cohomology of the mapping class groups. It follows that a discrete subgroup of a Lie group can never be realized as a subgroup of the mapping class groups.
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