Entropy, Shannon orbit equivalence, and sparse connectivity

We say that two free p.m.p. actions of countable groups are Shannon orbit equivalent if there is an equivalence between them whose associated cocycle partitions have finite entropy. show the acting sofic and each has a w-normal amenable subgroup which neither locally nor virtually cyclic then implies same maximum This extends result Austin beyond finitely generated setting consequence Bernoulli group with properties in question only they measure conjugate. Our arguments apply more generally to satisfying sparse connectivity condition we call property SC, yield entropy inequality under assumption one this property.