2007 Fiscal Year Final Research Report Summary
Research on the deformation space of hyperbolic structures on manifolds
| Project/Area Number |
18540080
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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 | Kyoto University |
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Principal Investigator |
FUJI Michihiko Kyoto University, Graduate School of Science, Associate Professor (60254231)
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| Co-Investigator(Kenkyū-buntansha) |
UE Masaaki Kyoto University, Graduate School of Science, Professor (80134443)
KAWAZUMI Nariya The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (30214646)
MORISHITA Masanori Kyushu University, Graduate School of Mathematical Sciences, Professor (40242515)
OCHIAI Hiroyuki Nagoya University, Graduate School of Mathematics, Professor (90214163)
MORIMOTO Yoshinori Kyoto University, Graduate School of Human and Environmental Studies, Professor (30115646)
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| Project Period (FY) |
2006 – 2007
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| Keywords | manifold / cone-structure / hyperbolic structure / deformation / differential equation / discrete group / automaton |
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| Research Abstract |
The main purpose of this research was to study deformations of a 3-dimensional hyperbolic cone manifold M with non-empty singular set. The head investigator Fujii and the investigator Ochiai constructed an algorithm for solving ordinary differential equations E of Fuchsian type which describe deformations of M. In fact, Fujii and Ochiai succeeded in solving the ordinary differential equations by making use of computers with performing the algorithm. This result was reported in the journal, Publ. Res.Inst Math. Sci 43 (2007). Fujii found some relation between the degeneration of hyperbolic structures on a surface and the confluence of deterministic singular points of ordinary differential equations E concretely. This result was reported in the joumal, Kyushu J.Math. 61 (2007). This result was reported in the workshop, "Actions of hyperbolic groups on specific manifolds and related topics", at Tokyo Metropolitan University, in June 2006. Furthermore, Fujii constructed a geodesic automatic structure for some discrete group G and Computed the growth function of G by making use of a computer program. This result was reported at tie conference, "Riemann surfaces and discrete groups", at Okayama University in January, 2008.
The investigator Ue researched the Fukumoto-Furuta invariant for Seifert 3-manifolds and the Neumann-Siebenmann invariant. Ue also studied its spin rational homology cobordism invariance. The investigator Morishita researched the analogy between 3-dimensional topology and Arithmetic. In particular, Morishita studied some relation between SL Chem-Simons theory and Hida-Mazur- theory. The investigator Kawazumi researched Riemann moduli space from the view points of differential geometry. In fact, Kawazumi found some relation between some Chem forms and Johnson homomorphisms for the mapping class groups.
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