Completely positive entropy actions of sofic groups with $\mathbb{Z}$ in their center

Let $Γ$ be a sofic group with a copy of $\mathbb{Z}$ in its center. We construct an uncountable family of pairwise nonisomorphic measure-preserving $Γ$ actions with completely positive entropy, none of which is a factor of a Bernoulli shift. Our construction shows that the relation of isomorphism among completely positive entropy $Γ$ actions is not smooth, in contrast with the relation of isomorphism among Bernoulli shifts.

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