Equivalence of Random Generics Is Not Essentially Free

Abstract

Let $M$ be a countable transitive model of a sufficiently large finite fragment of ZFC, and let $E_M^{\mathbb{B}}$ be equivalence of random generics over $M$. Smythe proved that $E_M^{\mathbb{B}}$ is not Borel-reducible to the orbit relation of a free action of any countable group belonging to $M$, and asked whether the restriction on the target group can be removed. We prove that if $A$ is a positive-measure Borel set of random reals over $M$ and $F$ is an essentially free countable Borel equivalence relation, then every Borel homomorphism from $E_M^{\mathbb{B}}\upharpoonright A$ to $F$ maps a conull subset of $A$ into a single $F$-class. Hence no positive-measure restriction of $E_M^{\mathbb{B}}$ is essentially free, or even weakly Borel-reducible to an essentially free relation. The proof combines Thomas's consequence of Popa cocycle superrigidity with a perfect family of $2$-marked groups and a Fubini argument using two successive random reals.