{
  "id": "2305.03927",
  "version": "v1",
  "published": "2023-05-06T04:50:23.000Z",
  "updated": "2023-05-06T04:50:23.000Z",
  "title": "The Borel complexity of the space of left-orderings, low-dimensional topology, and dynamics",
  "authors": [
    "Filippo Calderoni",
    "Adam Clay"
  ],
  "comment": "23 pages",
  "categories": [
    "math.LO",
    "math.GR",
    "math.GT"
  ],
  "abstract": "We find new criteria to analyze the complexity of the conjugacy equivalence relation $E_\\mathsf{lo}(G)$ for a given left-orderable group $G$. We show that $E_\\mathsf{lo}(G)$ is universal whenever $G$ is the free product of left-orderable groups, and that $E_\\mathsf{lo}(G)$ is not smooth whenever $G$ is simple and not bi-orderable, or when $G$ admits a closed $G$-invariant family of non-Conradian orderings. We apply these new criteria to study the smoothness of $E_\\mathsf{lo}(G)$ when $G$ is the fundamental group of $3$-manifold, and by using tools related to the L-space conjecture we show that if $G$ is a non-cyclic knot group, then $E_\\mathsf{lo}(G)$ is not smooth.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-06T04:50:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "borel complexity",
      "low-dimensional topology",
      "left-orderings",
      "conjugacy equivalence relation",
      "left-orderable group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}