Superrigidity, generalized harmonic maps and uniformly convex spaces

We prove several superrigidity results for isometric actions on metric spaces satisfying some convexity properties. First, we extend some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups. Second, we include the proof of an unpublished result on commensurability superrigidity due to Margulis. The proofs rely on certain notions of harmonic maps and the study of their existence, uniqueness, and continuity.