Stability in orbit equivalence for Baumslag-Solitar groups and Vaes groups

A measure-preserving action of a discrete countable group on a standard probability space is called stable if the associated equivalence relation is isomorphic to its direct product with the ergodic hyperfinite equivalence relation of type II1. We show that any Baumslag-Solitar group has such an ergodic, free and stable action. It follows that any Baumslag-Solitar group is measure equivalent to its direct product with any amenable group. The same property is obtained for the inner amenable groups of Vaes.