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We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of local and doubly empirical convergence developed by Austin we prove that in many cases the outer Pinsker factor of a product action is the product of the outer Pinsker factors. Our results are parallel to those of Seward for Rokhlin entropy. We use these Pinsker product formulas to show that if X is a compact group, and G is a sofic group with G↷X by automorphisms, then the outer Pinsker factor of G↷(X,mX) is given as a quotient by a G-invariant, closed, normal subgroup of X. We use our results to show that if G is sofic and f∈Mn(Z(G)) is invertible as a convolution operator l2(G)⊕n→l2(G)⊕n, then the action of G on the Pontryagin dual of Z(G)⊕n/Z(G)⊕nf has completely positive measure-theoretic entropy with respect to the Haar measure.
Classification :
37-XX, 22-XX
Mots-clés : Sofic groups, Pinsker factors, completely positive entropy, relative sofic entropy
Mots-clés : Sofic groups, Pinsker factors, completely positive entropy, relative sofic entropy
Affiliations des auteurs :
Ben Hayes 1
Ben Hayes. Relative entropy and the Pinsker product formula for sofic groups. Groups, geometry, and dynamics, Tome 15 (2021) no. 2, pp. 413-463. doi: 10.4171/ggd/602
@article{10_4171_ggd_602,
author = {Ben Hayes},
title = {Relative entropy and the {Pinsker} product formula for sofic groups},
journal = {Groups, geometry, and dynamics},
pages = {413--463},
year = {2021},
volume = {15},
number = {2},
doi = {10.4171/ggd/602},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/602/}
}
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