Cocycle superrigidity from higher rank lattices to $\mathrm{O}\mathrm{u}\mathrm{t}(F_N)$

We prove a rigidity result for cocycles from higher rank lattices to $\mathrm{O}\mathrm{u}\mathrm{t}(F_N)$ 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 $G\curvearrowright X$ 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 $G\times X\to$\mathrm{O}\mathrm{u}\mathrm{t}$(H)$ is cohomologous to a cocycle with values in a finite subgroup of $\mathrm{O}\mathrm{u}\mathrm{t}(H)$. This provides a dynamical version of theorems of Farb–Kaimanovich–Masur and Bridson–Wade asserting that every morphism 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.