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

TL;DR

This paper introduces a new algebraic invariant for measure-preserving equivalence relations and uses it to establish measure equivalence rigidity results for products of Baumslag-Solitar groups, also classifying their relative profinite completions.

Contribution

It presents a novel invariant for aperiodic inclusions and applies it to prove rigidity phenomena and classify products of Baumslag-Solitar groups and their completions.

Findings

Stable orbit equivalences preserve the number and isomorphism class of factors.

Complete classification of direct products of relative profinite completions of Baumslag-Solitar groups.

New measure equivalence rigidity phenomena for Baumslag-Solitar groups.

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.