Invariants of orbit equivalence relations and Baumslag-Solitar groups

TL;DR

This paper studies invariants of orbit equivalence relations for actions of Baumslag-Solitar groups, introducing a flow as an invariant and exploring measure equivalence with related groups.

Contribution

It introduces a flow invariant for ergodic actions of Baumslag-Solitar groups and extends results to measure equivalent groups.

Findings

Flow isomorphism class is an invariant under weak orbit equivalence.

Results connect orbit equivalence invariants with measure equivalence.

Provides new tools for classifying actions of Baumslag-Solitar groups.

Abstract

To an ergodic, essentially free and measure-preserving action of a non-amenable Baumslag-Solitar group on a standard probability space, a flow is associated. The isomorphism class of the flow is shown to be an invariant of such actions of Baumslag-Solitar groups under weak orbit equivalence. Results on groups which are measure equivalent to Baumslag-Solitar groups are also provided.