{
  "id": "2103.06663",
  "version": "v1",
  "published": "2021-03-11T13:40:26.000Z",
  "updated": "2021-03-11T13:40:26.000Z",
  "title": "Graph and wreath products in topological full groups of full shifts",
  "authors": [
    "Ville Salo"
  ],
  "comment": "10 pages",
  "categories": [
    "math.GR",
    "math.DS"
  ],
  "abstract": "We prove that the topological full group $[[X]]$ of a two-sided full shift $X = \\Sigma^{\\mathbb{Z}}$ contains every right-angled Artin group (also called a graph group). More generally, we show that the family of subgroups with \"linear look-ahead\" is closed under graph products. We show that the lamplighter group $\\mathbb{Z}_2 \\wr \\mathbb{Z}$ embeds in $[[X]]$, and conjecture that it does not embed in $[[X]]$ with linear look-ahead. Generalizing the lamplighter group, we show that whenever $G$ acts with \"unique moves\" (or at least \"move-$A$ithfully\"), we have $A \\wr G \\leq [[X]]$ for finite abelian groups $A$. We show that free products of finite and cyclic groups act with unique moves. We show that $\\mathbb{Z}^2$ does not admit move-$A$ithful actions, and conjecture that $\\mathbb{Z}_2 \\wr \\mathbb{Z}^2$ does not embed in $[[X]]$ at all. We show that topological full groups of all infinite nonwandering sofic shifts have the same subgroups, and that this set of groups is closed under commensurability. The group $[[X]]$ embeds in the higher-dimensional Thompson group $2$V, so it follows that $2$V contains all RAAGs, refuting a conjecture of Belk, Bleak and Matucci.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-11T13:40:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological full group",
      "full shift",
      "wreath products",
      "linear look-ahead",
      "lamplighter group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}