COCYCLE SUPERRIGIDITY FOR TRANSLATION ACTIONS OF PRODUCT GROUPS

Damien Gaboriau (1) , Adrian Ioana (2) , Robin Tucker-Drob (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 Mathematics Department; University of California San Diego
3 TAMU - Department of Mathematics [Texas]

Let G be either a profinite or a connected compact group, and Γ, Λ be finitely generated dense subgroups. Assuming that the left translation action of Γ on G is strongly ergodic, we prove that any cocycle for the left-right translation action of Γ × Λ on G with values in a countable group is " virtually " cohomologous to a group homomorphism. Moreover , we prove that the same holds if G is a (not necessarily compact) connected simple Lie group provided that Λ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE-and W *-superrigidity. In particular, we obtain the first examples of compact actions of F2 × F2 which are W *-superrigid.

Domaines: Systèmes dynamiques [math.DS]; Algèbres d'opérateurs [math.OA]

Damien Gaboriau, Adrian Ioana, Robin Tucker-Drob. COCYCLE SUPERRIGIDITY FOR TRANSLATION ACTIONS OF PRODUCT GROUPS. American Journal of Mathematics, In press, 141 (5), pp.1347-1374. ⟨hal-01299575⟩