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arXiv · 1603.07616

Cocycle superrigidity for translation actions of product groups

Damien Gaboriau·Adrian Ioana·Robin Tucker-Drob
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Abstract

Let $G$ be either a profinite or a connected compact group, and $\Gamma, \Lambda$ be finitely generated dense subgroups. Assuming that the left translation action of $\Gamma$ on $G$ is strongly ergodic, we prove that any cocycle for the left-right translation action of $\Gamma\times\Lambda$ 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 $\Lambda$ 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 $\mathbb F_2\times\mathbb F_2$ which are W$^*$-superrigid.

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BibTeXRIS

Damien Gaboriau, Adrian Ioana, Robin Tucker-Drob. 2016-03-24. Cocycle superrigidity for translation actions of product groups. https://arxiv.org/abs/1603.07616

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