Cocycle superrigidity for translation actions of product groups

DOI10.1353/AJM.2019.0035zbMATH Open1459.22006arXiv1603.07616OpenAlexW2305411190MaRDI QIDQ5236961

Damien Gaboriau, Adrian Ioana, Robin Tucker-Drob

Publication date: 16 October 2019

Published in: American Journal of Mathematics (Search for Journal in Brave)

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 $GammaimesLambda$ 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 $mathbb{F}_{2}imesmathbb{F}_2$ which are W${*}^{}$-superrigid.

Full work available at URL: https://arxiv.org/abs/1603.07616

zbMATH Keywords

cocyclestrongly ergodic action

Mathematics Subject Classification ID

Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations (37A20) Measurable group actions (22F10)

Related Items (7)

Superrigidity for dense subgroups of Lie groups and their actions on homogeneous spaces ⋮ Orbit equivalence rigidity for product actions ⋮ Cocycle superrigidity for profinite actions of irreducible lattices ⋮ Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces ⋮ Continuous orbit equivalence rigidity for left-right wreath product actions ⋮ W∗-superrigidity for coinduced actions ⋮ Cocycle superrigidity for profinite actions of property (T) groups

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