Cocycle superrigidity from higher rank lattices to Out(F_N)
Résumé
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.
Domaines
Théorie des groupes [math.GR]
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