2005 Fiscal Year Final Research Report Summary
Combinatorial harmonic maps and a rigidity of discrete-group actions
| Project/Area Number |
15540056
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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, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90244409)
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| Co-Investigator(Kenkyū-buntansha) |
KOTANI Motoko Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50230024)
FUJIWARA Koji Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (60229078)
KANAI Masahiko Nagoya University, Graduate School of Math., Professor, 大学院・多元数理科学研究科, 教授 (70183035)
NAYATANI Shin Nagoya University, Graduate School of Math., Professor, 大学院・多元数理科学研究科, 教授 (70222180)
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| Project Period (FY) |
2003 – 2005
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| Keywords | harmonic map / discrete group / nonpositively curved space / rigidity / fixed-point property / simplicial complex |
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| Research Abstract |
We developed a theory of combinatorial harmonic maps and obtained a fixed-point theorem for isometric discrete-group actions on nonpositively curved metric spaces.
Suppose that a discrete group Γ acts on a simplicial complex X properly discontinuously and cofinitely. Let Γ acts isometrically on a nonpositiovely curved metric space Y. We defined an energy for maps from X into Y which are equivariant with respect to the actions of Γ on both spaces. A combinatorial harmonic map is defined to be a critical point of this energy functional Y. We gave a sufficient condition for the gradient flow of the energy functional to converge to a constant map. This means that the given action of Γ on Y admits a global fixed point. Here the condition is independent of the action of Γ on Y, we conclude that Γ has a fixed-point property for Y. Namely, every isometric action of Γ on Y admits a global fixed-point. If Y is a Hilbert space, then this fixed-point property is known to be equivalent to Kazhdan's property (T). The sufficient condition is expressed in terms of a spectral invariant of a finite graph, a link of a vertex of X. We introduced a local invariant of nonpositively curved metric space, and gave an estimate of the spectral invariant in terms of the eigenvalue of the Lapalacian of the link and this local invariant.
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