2004 Fiscal Year Final Research Report Summary
Geometric study of infinite discrete groups
| Project/Area Number |
14540055
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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 |
FUJIWARA Koji Tohoku University, Graduate school of Science, Associate Professor, 大学院・理学研究科, 助教授 (60229078)
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| Co-Investigator(Kenkyū-buntansha) |
SHIOYA Takashi Tohoku University, Graduate school of Science, Associate Professor, 大学院・理学研究科, 助教授 (90235507)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Discrete groups / CAT(0) spaces |
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| Research Abstract |
The goal of this project is to study several problems on infinite discrete groups from the view point of geometry.
"Geometric group theory" has its root in the pionear work by Gromov, and has been developed mainly in USA and Europe in the last 15 years or so.
It is a dynamic field where one can apply classical combinatorial group theory, hyperbolic geometry, low dimensional topology, and the theory of mapping class groups. Unfortunately, there has not been much activities of this field in Japan yet.
During the three years, we not only conducted our research, but also tried to put a foundation of the research activities of this field in Japan.
One of our research themes has been on isometric actions of group on CAT(0) spaces. The notion of "CAT(0) spaces" was introduced by Gromov to geodesic metric spaces as a generalization of complete, simply-connected Riemannian manifolds, called "Hadamard manifolds".
Given a discrete group G, it is important and useful to find a metric space X on which G acts on by isometries, properly. One classical example is the action of a lattice subgroup in a Lie group on its symmetric space.
It would be interesting to find such X of minimal dimensions. In this direction, there has been a work by Brady-Crisp. We developed their work and found an answer to their question, found new examples, and formulated further questions in the paper "Parabolic isometries of CAT(0) spaces and CAT(0) dimensions", Fujiwara, Koji ; Shioya, Takashi ; Yamagata, Saeko. Algebr.Geom.Topol.4(2004), 861-892
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