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

Abstract

Let $\Gamma$ be a sofic group with a copy of $\mathbb{Z}$ in its center. We construct an uncountable family of pairwise nonisomorphic measure-preserving $\Gamma$ 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 $\Gamma$ actions is not smooth, in contrast with the relation of isomorphism among Bernoulli shifts.