The f-invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen, who first used it to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only if their base measures have the same Shannon entropy. Bowen also showed that the f-invariant is a variant of sofic entropy; in particular, it is the exponential growth rate of the expected number of good models over a uniform random homomorphism. In this paper we present an analogous formula for the relative f-invariant and use it to prove a formula for the exponential growth rate of the expected number of good models over a random sofic approximation which is a type of stochastic block model.

We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.

We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if $\unicode[STIX]{x1D6E4}$ is a nonamenable group and $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.