Generalized harmonic maps and representations of discrete groups

This paper considers generalized harmonic maps from a simplicial complex to a complete metric space of (globally) non-positive curvature. It is proved that if a simplicial complex admits an " admissible weight" satisfying a l ocal combinatorial condition, then any such generalized harmonic maps must be constant maps. The local combinatorial condition is in terms of a nonlin ear generalization of the first eigenvalue of a graph. This has application s in the Archimedean and non-Archimedean representations of finitely presen table groups.