Generalized harmonic maps and representations of discrete groups

We study the representations of finitely presentable groups, or equivalently fundamental groups of finite simplicial complexes. We show that if a simplicial complex admits an "admissible weight" satisfying a certain combinatorial condition, then its fundamental group has Kazhdan's property (T). Moreover, any Zariski dense representation of its fundamental group into any real simple Lie group with trivial center must have bounded image. The combinatorial condition is an analogue of curvature condition on Riemannian manifolds. In particular, we show the Bruhat-Tit buildings associated with most simply connected p-adic simple Lie group of rank greater than or equal to two admit such admissible weight. Therefore the latter can viewed as a generalization of Margulis' superrigidity theorem for the Archimedean representations of cocompact lattices in such groups. We also show that certain similar combinatorial conditions imply the boundedness of non-Archimedean representations. The strategy we adopt involves generalized harmonic maps from simplicial complexes to Riemannian manifolds and Bruhat-Tits buildings.