We prove the rigid consequence of cocycles from higher-order lattices to $\mathrm{Out}(F_N)$ and more generally to the outer automorphism group of the untwisted hyperbolic group. More precisely, let $G$ be either the product of connected higher-order simple algebraic groups over local fields, or a lattice of such products. Let $G\curvearrowright X$ be an ergodic measure-conserving action on the standard probability space, and $H$ be the untwisted hyperbolic group. Prove that every Borel cocycle $G\times X\to\mathrm{Out}(H)$ is homologous to the valued cocycles of finite subgroups of $\mathrm{Out}(H)$ To do. This is the dynamic of the Farb-Kaimanovich-Masur and Bridson-Wade theorem, which asserts all morphisms from $G$ to either the mapping class group of finite type surfaces or the outer automorphism group of free groups. provide the version. I have a finite image.

The main new geometric tool is the centroid map, which associates a finite set of (relative) free splits to every triple point on the boundary of the (relative) free factor graph.