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We say that two free probability-measure-preserving actions of countablegroups are Shannon orbit equivalent if there is an orbit equivalence betweenthem whose associated cocycle partitions have finite Shannon entropy. We showthat if the acting groups are sofic and each has a w-normal amenable subgroupwhich is neither locally finite nor virtually cyclic then Shannon orbitequivalence implies that the actions have the same maximum sofic entropy. Thisextends a result of Austin beyond the finitely generated amenable setting andhas the consequence that two Bernoulli actions of a group with the propertiesin question are Shannon orbit equivalent if and only if they are measureconjugate. Our arguments apply more generally to actions satisfying a sparseconnectivity condition which we call property SC, and yield an entropyinequality under the assumption that one of the actions has this property.

Entropy, Shannon orbit equivalence, and sparse connectivity