{
  "id": "2011.14604",
  "version": "v1",
  "published": "2020-11-30T08:05:12.000Z",
  "updated": "2020-11-30T08:05:12.000Z",
  "title": "Factor of i.i.d. processes and Cayley diagrams",
  "authors": [
    "Riley Thornton"
  ],
  "categories": [
    "math.CO",
    "math.DS",
    "math.LO"
  ],
  "abstract": "We investigate when a $\\Gamma$-f.i.i.d. solution (or approximate solution) to a local labelling problem on a Cayley graph $G=\\operatorname{Cay}(\\Gamma, E)$ lifts to an $\\operatorname{Aut}(G)$-f.i.i.d. solution (or approximate solution). A Cayley diagram is an edge labelling of $G$ by elements of the generating set $E$ so that the cycles correspond exactly to paths labelled by relations in $\\Gamma$. We show that if $G$ admits an $\\operatorname{Aut}(G)$-f.i.i.d. (approximate) Cayley diagram, then any $\\Gamma$-f.i.i.d. (approximate) solution to a local combinatorial problem lifts to $\\operatorname{Aut}(G)$-f.i.i.d. (approximate) solution. We also establish a number of results on which graphs admit such a Cayley diagram. We show that Cayley graphs of amenable groups and free groups always admit $\\operatorname{Aut}(G)$-f.i.i.d. approximate Cayley diagrams, and that torsion-free nilpotent groups never admit nontrivial $\\operatorname{Aut}(G)$-f.i.i.d. Cayley diagrams. Finally, we construct a Cayley graph with a $\\Gamma$-f.i.i.d. 3-colorings that does not lift to even an approximate $\\operatorname{Aut}(G)$-f.i.i.d. 3-coloring. Our construction also answers a question of Weilacher.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-30T08:05:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A50",
      "03E15",
      "05C63"
    ],
    "keywords": [
      "cayley graph",
      "local combinatorial problem lifts",
      "approximate solution",
      "approximate cayley diagrams",
      "torsion-free nilpotent groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}