Cocycle superrigidity for translation actions of product groups

D. Gaboriau, A. Ioana, and R. Tucker-Drob

American Journal of Mathematics, 141(2019), 1347-1374.

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HAL: https://hal.archives-ouvertes.fr/hal-01299575
arxiv: http://arxiv.org/abs/1603.07616

Abstract

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.