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A probability-measure-preserving transformation has the Weak Pinsker Property(WPP) if for every $\epsilon>0$ it is measurably conjugate to the directproduct of a transformation with entropy $<\epsilon$ and a Bernoulli shift. Ina recent breakthrough, Tim Austin proved that every ergodic transformationsatisfies this property. Moreover, the natural analog for amenable groupactions is also true. By contrast, this paper provides a counterexample inwhich the group $\Gamma$ is a non-abelian free group and the notion of entropyis sofic entropy. The counterexample is a limit of hardcore models on randomregular graphs. In order to prove that it does not have the WPP, this paperintroduces new measure conjugacy invariants based on the growth of homology ofthe model spaces of the action. The main result is obtained by showing that anyaction with the WPP has subexponential homology growth in dimension $0$, whilethe counterexample has exponential homology growth in dimension $0$.

Sofic homological invariants and the Weak Pinsker Property