Combinatorial harmonic maps and a rigidity of discrete-group actions

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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.