COCYCLE SUPERRIGIDITY FROM HIGHER RANK LATTICES TO Out(FN)

We prove a rigidity result for cocycles from higher rank lattices to Out(FN) and more generally to the outer automorphism group of a torsion -free hyperbolic group. More precisely, let G be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let G X be an ergodic measure-preserving action on a standard probability space, and let H be a torsion-free hyperbolic group. We prove that every Borel cocycle G x X -> Out(H) is cohomologous to a cocycle with values in a finite subgroup of Out(H). This provides a dynamical version of theorems of Farb-Kaimanovich-Masur and Bridson-Wade asserting that every homomorphism from G to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image. The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.