Combinatorial harmonic maps and a rigidity of discrete-group actions

本文发展了组合调和映射理论，得到了非正曲度量空间上等距离散群作用的一个不动点定理.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。

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.