Generalized harmonic maps and representations of discrete groups

Abstract

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 local combinatorial condition, then any such generalized harmonic maps must be constant maps. The local combinatorial condition is in terms of a nonlinear generalization of the first eigenvalue of a graph. This has applications in the Archimedean and non-Archimedean representations of finitely presentable groups.