# Cocycle superrigidity for translation actions of product
groups

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

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

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Download: pdf (430K)

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 𝔽_{2}×𝔽_{2}
which are W^{∗}-superrigid.