{
  "id": "2103.14911",
  "version": "v1",
  "published": "2021-03-27T13:43:57.000Z",
  "updated": "2021-03-27T13:43:57.000Z",
  "title": "Subgroups of $\\mathrm{PL}_+ I$ which do not embed into Thompson's group $F$",
  "authors": [
    "James Hyde",
    "Justin Tatch Moore"
  ],
  "comment": "24 pages. Comments welcome",
  "categories": [
    "math.GR"
  ],
  "abstract": "We will give a general criterion - the existence of an $F$-obstruction - for showing that a subgroup of $\\mathrm{PL}_+ I$ does not embed into Thompson's group $F$. An immediate consequence is that Cleary's \"golden ratio\" group $F_\\tau$ does not embed into $F$. Our results also yield a new proof that Stein's groups $F_{p,q}$ do not embed into $F$, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of $F$-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the paper, we prove a dichotomy theorem for subgroups of $\\mathrm{PL}_+ I$. In addition to playing a central role in our proof, it is strong enough to imply both Rubin's Reconstruction Theorem restricted to the class of subgroups of $\\mathrm{PL}_+ I$ and also Brin's Ubiquity Theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-27T13:43:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "thompsons group",
      "brins ubiquity theorem",
      "rubins reconstruction theorem",
      "general criterion",
      "immediate consequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}