{
  "abstract": "We motivate and study the class $\\mathcal{C}$ of countable groups $G$ such that the conjugacy relation between minimal actions of $G$ on $\\mathbb{R}$ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal. No example of amenable group outside of $\\mathcal{C}$ is known. We show a number of stability properties of $\\mathcal{C}$ under group-theoretic operations and that $\\mathcal{C}$ contains all finitely generated groups of piecewise affine homeomorphisms of the interval. We exhibit a finitely generated group $G$ that is not in $\\mathcal{C}$, such that $G$ is amenable if and only if Thompson's group $F$ is amenable. We also prove that the semiconjugacy relation among cocompact actions of a countable group $G$ is smooth if and only if $G \\in \\mathcal{C}$, and that it is essentially countable even when $G$ is not finitely generated. In the Appendix, we show that there is no good analogue of the space of harmonic actions for a countable non-finitely generated group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.13406",
  "author_total": 3,
  "authors": [
    {
      "name": "Joaquín Brum"
    },
    {
      "name": "Martín Gilabert Vio"
    },
    {
      "name": "Nicolás Matte Bon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftkqhqk04rdbm.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "ftkqhqk04rdbm",
  "issns": [],
  "landing_url": "https://hal.science/hal-05852230v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05852230/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftkqhqk04rdbm.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:13Z",
      "native_id": "hal-05852230",
      "source": "hal",
      "url": "https://hal.science/hal-05852230v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Groups with classifiable actions on the line",
  "title_text": "Groups with classifiable actions on the line",
  "type": "preprint",
  "updated_at": "2026-11-06T12:20:44Z",
  "url": "https://research.draconitic.com/paper/ftkqhqk04rdbm",
  "venue": "HAL",
  "versions": []
}