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

Vincent Guirardel; Camille Horbez; Jean Lécureux

doi:10.3934/jmd.2022010

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2022, Volume 18: 291-344. Doi: 10.3934/jmd.2022010

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

1.
Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
2.
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, 91405, Orsay, France

Received: May 26, 2020

Revised: January 10, 2022

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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 homomorphism 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.

Keywords:

Superrigidity,
cocycles,
lattices in higher rank algebraic groups,
outer automorphism groups of free groups and hyperbolic groups.

Mathematics Subject Classification: Primary: 37A20; Secondary: 20E36, 20F65, 20F67, 20G25, 22E40.

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