The measure equivalence classification of Baumslag-Solitar groups
Abstract
In this article we show that all the nonamenable Baumslag-Solitar groups with nonunimodular Bass-Serre tree, i.e., BS(p,q) with |p|,|q| ≠ 1 and |p|≠ |q|, are measure equivalent to each other, thereby completing the measure equivalence classification of Baumslag-Solitar groups: each 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 Fn × Z (|p|=|q|≠ 1), or is nonamenable with nonunimodular Bass-Serre tree (|p|,|q| ≠ 1 and |p| ≠ |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 III equivalence relations and graphings of independent interest. For example, we show that every ergodic type III0 graphing admits an ergodic hyperfinite subgraph for which the natural factor map of Krieger flows is an isomorphism.
Related
- On Small Doubling in Right-Ordered Groups and Baumslag-Solitar Groups-II
- On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and SLn(Z)
- Singular Loci of SL2(C)-Representation Varieties of Non-Hopfian Baumslag--Solitar Groups
- Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions
- The Complexity of Conjugacy, Orbit Equivalence, and von Neumann Equivalence of Actions of Nonamenable Groups
BibTeX
@misc{gaboriau2026measure,
title = {{The measure equivalence classification of Baumslag-Solitar groups}},
author = {Damien Gaboriau and Antoine Poulin and Anush Tserunyan and Robin Tucker-Drob and Konrad Wróbel},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34323},
eprint = {2610.34323},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05843185v1},
}
RIS
TY - UNPB
TI - The measure equivalence classification of Baumslag-Solitar groups
AU - Damien Gaboriau
AU - Antoine Poulin
AU - Anush Tserunyan
AU - Robin Tucker-Drob
AU - Konrad Wróbel
PY - 2026
DA - 2026/10/25
PB - HAL
DO - 10.48550/arxiv.2610.34323
UR - https://hal.science/hal-05843185v1
AB - In this article we show that all the nonamenable Baumslag-Solitar groups with nonunimodular Bass-Serre tree, i.e., BS(p,q) with |p|,|q| ≠ 1 and |p|≠ |q|, are measure equivalent to each other, thereby completing the measure equivalence classification of Baumslag-Solitar groups: each 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 F_n × Z (|p|=|q|≠ 1), or is nonamenable with nonunimodular Bass-Serre tree (|p|,|q| ≠ 1 and |p| ≠ |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 III equivalence relations and graphings of independent interest. For example, we show that every ergodic type III_0 graphing admits an ergodic hyperfinite subgraph for which the natural factor map of Krieger flows is an isomorphism.
ER -