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

Abstract

We introduce an algebraic invariant for aperiodic inclusions of probability measure preserving 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 generalises some of the results obtained by Kida 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.

23 pages

References in corpus (5)

• A survey of Measured Group Theory
• Measure equivalence rigidity of the mapping class group
• Invariants of orbit equivalence relations and Baumslag-Solitar groups
• Examples of amalgamated free products and coupling rigidity
• Partial classification of the Baumslag-Solitar group von Neumann algebras

Cited by in corpus (1)

• Measure equivalence classification of transvection-free right-angled Artin groups