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Motivated by our previous results, we investigate structural properties ofprobability measure-preserving actions of sofic groups relative to theirPinsker factor. We also consider the same properties relative to the OuterPinsker factor, which is another generalization of the Pinsker factor in thenonamenable case. The Outer Pinsker factor is motivated by extension entropyfor extensions, which fixes some of the "pathological" behavior of soficentropy: namely increase of entropy under factor maps. We show that anarbitrary probability measure-preserving action of a sofic group is mixingrelative to its Pinsker and Outer Pinsker factors and, if the group isnonamenable, it has spectral gap relative to its Pinsker and Outer Pinskerfactors. Our methods are similar to those we developed in "Polish models andsofic entropy" and based on representation-theoretic techniques. One crucialdifference is that instead of considering unitary representations of a group wemust consider *-representations of algebraic crossed products.

Mixing and Spectral Gap Relative to Pinsker Factors for Sofic Groups