Consider a compact group $G$ together with two countable dense subgroups $\Gamma$ and $\Lambda$. I will discuss a result which asserts that, under certain mild assumptions, any cocycle for the left-right translation action of $\Gamma\times\Lambda$ of $G$ with values in a countable group is (virtually) cohomologous to a homomorphism. This is joint work with D. Gaboriau and R. Tucker-Drob.
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