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Cocycle superrigidity from higher rank lattices to Out(F_N)

Abstract : We prove a rigidity result for cocycles from higher rank lattices to 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 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 × X → Out(H) is cohomologous to a cocycle with values in a finite subgroup of 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.
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https://hal.archives-ouvertes.fr/hal-02611060
Contributor : Vincent Guirardel <>
Submitted on : Wednesday, November 18, 2020 - 2:23:59 PM
Last modification on : Friday, November 20, 2020 - 3:34:52 AM

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  • HAL Id : hal-02611060, version 1

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Vincent Guirardel, Camille Horbez, Jean Lécureux. Cocycle superrigidity from higher rank lattices to Out(F_N). 2020. ⟨hal-02611060⟩

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