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

TL;DR

This paper establishes a cocycle superrigidity result for higher rank lattices acting on the outer automorphism groups of hyperbolic groups, showing such cocycles are essentially finite and extending known rigidity theorems.

Contribution

It introduces a new geometric barycenter map and proves a superrigidity theorem for cocycles into Out(H), generalizing previous results for mapping class groups.

Findings

Cocycles from higher rank lattices to Out(H) are cohomologous to finite subgroup-valued cocycles.

The paper extends rigidity results to actions on the outer automorphism groups of hyperbolic groups.

A new barycenter map for boundary triples in free factor graphs is developed.

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.