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

Abstract

We prove a rigidity result for cocycles from higher rank lattices to $\mathrm{Out}(F_N)$ 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 $G\times X\to \mathrm{Out}(H)$ is cohomologous to a cocycle with values in a finite subgroup of $\mathrm{Out}(H)$. 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.

arXiv commentv2: Accepted in the Journal of Modern Dynamics. This version is an authors copy of the accepted manuscript; the version of record, in its final form, can be found at https://www.aimsciences.org/article/doi/10.3934/jmd.2022010

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