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We introduce an algebraic invariant for aperiodic inclusions of probabilitymeasure preserving equivalence relations. We use this invariant to prove thatevery stable orbit equivalence between free pmp actions of direct products ofnon-amenable Baumslag-Solitar groups whose canonical subgroup actsaperiodically forces the number of factors of the products to be the same andthe factors to be isomorphic after permutation. This generalises some of theresults obtained by Kida and moreover provides new measure equivalence rigidityphenomena for Baumslag-Solitar groups. We also obtain a complete classificationof direct products of relative profinite completions of Baumslag-Solitargroups, continuing recent work of Elder and Willis.

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