Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 12 min ago
PreprintCC BY 4.0arXiv

Classification complexity of homeomorphism group actions

  • Michal Hevessy
  • Benjamin Vejnar
Open at sourcePDFDOICite

Abstract

In this paper, we study how the classification complexity of natural orbit equivalence relations changes when the full homeomorphism group of a compact metrizable space is replaced by a dense non-closed subgroup. For a compact space X and a subgroup G ≤ ℋ(X), we consider three canonical actions: the left shift action on ℋ(X), the induced hyperspace action on ℱ(X), and the conjugation action on G We first analyze subgroups of the group ℋ+([0,1]) of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.

Related

  1. On the homeomorphism group of the Cantor set: monotone and measure preserving maps
    Benjamin Vejnar
    arXivPreprintCC BY 4.0Full textDynamical systemsarXiv
  2. The Complexity of Conjugacy, Orbit Equivalence, and von Neumann Equivalence of Actions of Nonamenable Groups
    Eusebio Gardella, Martino Lupini
    Memoirs of the American Mathematical SocietyPeer-reviewedLicence not statedFull text6 citedMathematical physicsOpenAlex
  3. Equidistribution of currents under Anosov group actions
    Nguyen-Bac Dang, Michael Kapovich, Mikhail Lyubich +1
    HALPreprintLicence not statedGeneral mathematicsHAL
  4. Some homotopy coherent finite group actions on spheres
    Manuel Krannich, Oscar Randal-Williams
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  5. Finite group actions on nonpositively curved 3-manifolds
    Bena Tshishiku, Zhengyu Zou
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{hevessy2026classification,
  title = {{Classification complexity of homeomorphism group actions}},
  author = {Michal Hevessy and Benjamin Vejnar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.30346},
  eprint = {2608.30346},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30346},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Classification complexity of homeomorphism group actions
AU  - Michal Hevessy
AU  - Benjamin Vejnar
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2608.30346
UR  - https://arxiv.org/abs/2608.30346
AB  - In this paper, we study how the classification complexity of natural orbit equivalence relations changes when the full homeomorphism group of a compact metrizable space is replaced by a dense non-closed subgroup. For a compact space X and a subgroup G ≤ ℋ(X), we consider three canonical actions: the left shift action on ℋ(X), the induced hyperspace action on ℱ(X), and the conjugation action on G We first analyze subgroups of the group ℋ^+([0,1]) of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.
ER  -