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

TL;DR

The paper constructs many non-isomorphic measure-preserving actions of sofic groups with a central $\

Contribution

It introduces a new family of completely positive entropy actions for sofic groups with a central $\

Findings

Uncountably many non-isomorphic actions with completely positive entropy.

These actions are not factors of Bernoulli shifts.

The isomorphism relation among these actions is not smooth.

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.