The paper concerns superrigidity results for homomorphisms from lattices and cocycles over ergodic actions ranging into convergence groups, including Gromov hyperbolic and relatively hyperbolic group pairs. The source groups include higher rank algebraic groups, products of two or more general non-amenable groups, groups acting transitively on $\tilde{A}_2$ buildings. Commensurator superrigidity is also proved.
The approach is based on a notion of a generalized Weyl group, which reflects the (higher) rank assumption of the acting group. We also prove a useful cocycle reduction lemma. These tools allow to give a uniform proof for a variety of known superrigidity results and to prove new ones.