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PreprintCC BY 4.0HAL

Groups with classifiable actions on the line

  • Joaquín Brum
  • Martín Gilabert Vio
  • Nicolás Matte Bon
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Abstract

We motivate and study the class 𝒞 of countable groups G such that the conjugacy relation between minimal actions of G on ℝ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal. No example of amenable group outside of 𝒞 is known. We show a number of stability properties of 𝒞 under group-theoretic operations and that 𝒞 contains all finitely generated groups of piecewise affine homeomorphisms of the interval. We exhibit a finitely generated group G that is not in 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 ∈ 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.

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Cite

BibTeXRIS
BibTeX
@misc{brum2026groups,
  title = {{Groups with classifiable actions on the line}},
  author = {Joaquín Brum and Martín Gilabert Vio and Nicolás Matte Bon},
  howpublished = {HAL},
  year = {2026},
  month = nov,
  eprint = {2605.13406},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05852230v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Groups with classifiable actions on the line
AU  - Joaquín Brum
AU  - Martín Gilabert Vio
AU  - Nicolás Matte Bon
PY  - 2026
DA  - 2026/11/04
PB  - HAL
UR  - https://hal.science/hal-05852230v1
AB  - We motivate and study the class 𝒞 of countable groups G such that the conjugacy relation between minimal actions of G on ℝ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal. No example of amenable group outside of 𝒞 is known. We show a number of stability properties of 𝒞 under group-theoretic operations and that 𝒞 contains all finitely generated groups of piecewise affine homeomorphisms of the interval. We exhibit a finitely generated group G that is not in 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 ∈ 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.
ER  -