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

Abstract

Let BS(n 1 , m 1 ) X 1 and BS(n 2 , m 2 ) 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 II 1 factors forces BS(n 1 , m 1 ) ∼ = BS(n 2 , m 2 ) when |n 1 | = |m 1 | and BS(n 1 , m 1 ) ∼ = BS(n 2 , ±m 2 ) when |n 1 | = |m 1 |. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum in [HR13].