The measure equivalence classification of Baumslag-Solitar groups

In this article we show that all the nonamenable Baumslag-Solitar groups with nonunimodular Bass-Serre tree, i.e., $\mathrm{BS}(p,q)$ with $|p|,|q| \neq 1$ and $|p|\neq |q|$, are measure equivalent to each other, thereby completing the measure equivalence classification of Baumslag-Solitar groups: each $\mathrm{BS}(p,q)$ belongs to one of three measure equivalence classes according to whether it is amenable ($|p|=1$ or $|q|=1$), virtually isomorphic to $\mathbf{F}_n \times \mathbf{Z}$ ($|p|=|q|\neq 1$), or is nonamenable with nonunimodular Bass-Serre tree ($|p|,|q| \neq 1$ and $|p| \neq |q|$). More generally, we obtain the similar classification of the generalized Baumslag-Solitar groups of rank $1$ (i.e., groups that act cocompactly on a tree with edge and vertex stabilizers infinite cyclic). Our proof also establishes a classification of all treeable lcsc groups into exactly six measure equivalence classes. Our new contribution which completes this taxonomy is: all nonunimodular nonamenable treeable lcsc groups are measure equivalent. Along the way we obtain several results about type $\mathrm{III}$ equivalence relations and graphings of independent interest. For example, we show that every ergodic type $\mathrm{III}_0$ graphing admits an ergodic hyperfinite subgraph for which the natural factor map of Krieger flows is an isomorphism.