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Combinatorial harmonic maps and a rigidity of discrete-group actions

组合调和映射和离散群作用的刚性

基本信息

批准号：
15540056
负责人：
IZEKI Hiroyasu
金额：
$ 2.37万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540056/
关键词：
harmonic map discrete group nonpositively curved space rigidity fixed-point property simplicial complex

项目摘要

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.

我们发展了组合调和映射理论，并获得了非正曲度量空间上等距离散群作用的不动点定理。假设离散群Γ适当不连续且上有限地作用在单纯复形X上。设Γ等距作用在非正曲度量空间Y上.我们定义了从X到Y的映射的能量，这些映射关于Γ在两个空间上的作用是等变的。定义了一个组合调和映射作为该能量泛函Y的临界点。给出了能量泛函的梯度流收敛于常数映射的一个充分条件。这意味着给定的Γ在Y上的作用允许一个全局不动点。这里的条件与Γ在Y上的作用无关，我们得出结论：Γ对于Y具有不动点性质。也就是说，Γ在Y上的每一个等距作用都有一个整体不动点。如果Y是一个希尔伯特空间，那么这个不动点性质已知等价于Kazhdan的性质（T）。该充分条件用有限图的谱不变量表示，即X的一个顶点的链。本文引入了非正曲度量空间的一个局部不变量，并利用链的Lapalacian特征值和这个局部不变量给出了谱不变量的一个估计。