{
  "id": "1105.4017",
  "version": "v3",
  "published": "2011-05-20T06:17:53.000Z",
  "updated": "2014-01-08T18:25:12.000Z",
  "title": "Schreier graphs of actions of Thompson's group F on the unit interval and on the Cantor set",
  "authors": [
    "Dmytro Savchuk"
  ],
  "comment": "19 pages; 9 figures; final version to appear in Geometriae Dedicata; referee's comments and suggestions were incorporated; references updated",
  "categories": [
    "math.GR"
  ],
  "abstract": "Schreier graphs of the actions of Thompson's group $F$ on the orbits of all points of the unit interval and of the Cantor set with respect to the standard generating set $\\{x_0,x_1\\}$ are explicitly constructed. The closure of the space of pointed Schreier graphs of the action of $F$ on the orbits of dyadic rational numbers and corresponding Schreier dynamical system are described. In particular, we answer the question of Grigorchuk on the Cantor-Bendixson rank of the underlying space of the Schreier dynamical system in the context of $F$. As applications we prove that the pointed Schreier graphs of points from $(0,1)$ are amenable, have infinitely many ends, and are pairwise non-isomorphic. Moreover, we prove that points $x,y\\in(0,1)$ have isomorphic non-pointed Schreier graphs if and only if they belong to the same orbit of $F$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-01-08T18:25:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "05C25"
    ],
    "keywords": [
      "thompsons group",
      "unit interval",
      "cantor set",
      "schreier dynamical system",
      "dyadic rational numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.4017S"
    }
  }
}