Stability in orbit equivalence for Baumslag-Solitar groups and Vaes groups

TL;DR

This paper proves that Baumslag-Solitar groups and certain Vaes groups possess stable measure-preserving actions, implying their measure equivalence to their product with any amenable group, advancing understanding of their orbit equivalence properties.

Contribution

It establishes the existence of stable actions for Baumslag-Solitar groups and extends this property to inner amenable groups of Vaes, revealing new measure equivalence relations.

Findings

Baumslag-Solitar groups have ergodic, free, stable actions.

Such groups are measure equivalent to their product with any amenable group.

The property extends to inner amenable groups of Vaes.

Abstract

A measure-preserving action of a discrete countable group on a standard probability space is called stable if the associated equivalence relation is isomorphic to its direct product with the ergodic hyperfinite equivalence relation of type II_1. We show that any Baumslag-Solitar group has such an ergodic, free and stable action. It follows that any Baumslag-Solitar group is measure equivalent to its direct product with any amenable group. The same property is obtained for the inner amenable groups of Vaes.