Cocycle superrigidity from higher rank lattices to $ {{\rm{Out}}}{(F_N)} $

We prove a rigidity result for cocycles from higher rank lattices to \begin{document}$ \mathrm{Out}(F_N) $\end{document} and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let \begin{document}$ G $\end{document} be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let \begin{document}$ G \curvearrowright X $\end{document} be an ergodic measure-preserving action on a standard probability space, and let \begin{document}$ H $\end{document} be a torsion-free hyperbolic group. We prove that every Borel cocycle \begin{document}$ G\times X\to \mathrm{Out}(H) $\end{document} is cohomologous to a cocycle with values in a finite subgroup of \begin{document}$ \mathrm{Out}(H) $\end{document}. This provides a dynamical version of theorems of Farb–Kaimanovich–Masur and Bridson–Wade asserting that every homomorphism from \begin{document}$ G $\end{document} 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.