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

TL;DR

This paper establishes a rigidity result for crossed product II_1 factors arising from ergodic actions of Baumslag-Solitar groups, showing that isomorphisms imply group isomorphisms under certain conditions.

Contribution

It improves previous orbit equivalence rigidity results by proving that isomorphisms of crossed product factors determine the Baumslag-Solitar groups involved.

Findings

Isomorphism of crossed products implies group isomorphism for certain Baumslag-Solitar groups.

The result distinguishes groups based on the absolute values of parameters n and m.

The theorem refines previous rigidity results by Houdayer and Raum.

Abstract

Let BS(n_1,m_1) $\curvearrowright$ X_1 and BS(n_2,m_2) $\curvearrowright$ 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) $\cong$ BS(n_2,m_2) when |n_1| $\neq$ |m_1| and BS(n_1,m_1) $\cong$ BS(n_2,$\pm$m_2) when |n_1| = |m_1|. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum.