A rigidity result for crossed products of actions of Baumslag-Solitar groups

Let BS(n(1), m(1)) curved right arrow X-1 and BS(n(2), m(2)) curved right arrow X-2 be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag-Solitar groups whose canonical almost normal abelian subgroups act aperiodically. We prove that an isomorphism between the corresponding crossed product II1 factors forces BS(n(1), m(1)) congruent to BS(n(2), m(2)) when vertical bar n(1)vertical bar not equal vertical bar m(1)vertical bar and BS(n(1), m(1)) congruent to BS(n(2), +/- m(2)) when vertical bar n(1)vertical bar = vertical bar m(1)vertical bar. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum in [Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity, J. Topol. 8 (2015) 295-313].