Cocycle superrigidity for median spaces of finite rank

We investigate cocycle superrigidity in the setting of finite rank median spaces. By exploiting a dynamical approach to superrigidity, we establish, for a median space X of finite rank, the superrigidity of Isom(X)-valued cocycles for a product of locally compact second countable groups. In the case of action by irreducible lattices in such product groups, this approach yields a new and self-contained proof of the superrigidity of homomorphisms.