Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity

We introduce an algebraic invariant for aperiodic inclusions of probability measure preserving (pmp) equivalence relations. We use this invariant to prove that every stable orbit equivalence between free pmp actions of direct products of non-amenable Baumslag-Solitar groups whose canonical subgroup acts aperiodically, forces the number of factors of the products to be the same and the factors to be isomorphic after permutation. This generalizes some of the results obtained by Kida ['Invariants of orbit equivalence relations and Baumslag-Solitar groups', Tohoku Math. J. (2) 66 (2014) 205-258] and, moreover, provides new measure equivalence rigidity phenomena for Baumslag-Solitar groups. We also obtain a complete classification of direct products of relative profinite completions of Baumslag-Solitar groups, continuing recent work of Elder and Willis ['Totally disconnected groups from Baumslag-Solitar groups', Preprint, 2013, arXiv:1301.4775].