{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34323",
  "author_total": 5,
  "authors": [
    {
      "name": "Damien Gaboriau"
    },
    {
      "name": "Antoine Poulin"
    },
    {
      "name": "Anush Tserunyan"
    },
    {
      "name": "Robin Tucker-Drob"
    },
    {
      "name": "Konrad Wróbel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/es3y8m8pgaxm9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34323",
  "fulltext_url": null,
  "id": "es3y8m8pgaxm9",
  "issns": [],
  "landing_url": "https://hal.science/hal-05843185v1",
  "language": "en",
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7223980444",
  "pdf_url": "https://arxiv.org/pdf/2610.34323",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/es3y8m8pgaxm9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:03Z",
      "native_id": "hal-05843185",
      "source": "hal",
      "url": "https://hal.science/hal-05843185v1"
    },
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34323",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34323"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "The measure equivalence classification of Baumslag-Solitar groups",
  "title_text": "The measure equivalence classification of Baumslag-Solitar groups",
  "type": "preprint",
  "updated_at": "2026-10-31T12:20:56Z",
  "url": "https://research.draconitic.com/paper/es3y8m8pgaxm9",
  "venue": "HAL",
  "versions": []
}