{
  "id": "1407.7745",
  "version": "v2",
  "published": "2014-07-29T14:53:12.000Z",
  "updated": "2014-12-03T20:30:50.000Z",
  "title": "Groups with Context-Free Co-Word Problem and Embeddings into Thompson's Group $V$",
  "authors": [
    "Rose Berns-Zieve",
    "Dana Fry",
    "Johnny Gillings",
    "Hannah Hoganson",
    "Heather Mathews"
  ],
  "comment": "13 pages, 8 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a finitely generated group, and let $\\Sigma$ be a finite subset that generates $G$ as a monoid. The \\emph{word problem of $G$ with respect to $\\Sigma$} consists of all words in the free monoid $\\Sigma^{\\ast}$ that are equal to the identity in $G$. The \\emph{co-word problem of $G$ with respect to $\\Sigma$} is the complement in $\\Sigma^{\\ast}$ of the word problem. We say that a group $G$ is \\emph{co$\\mathcal{CF}$} if its co-word problem with respect to some (equivalently, any) finite generating set $\\Sigma$ is a context-free language. We describe a generalized Thompson group $V_{(G, \\theta)}$ for each finite group G and homomorphism $\\theta$: $G \\rightarrow G$. Our group is constructed using the cloning systems introduced by Witzel and Zaremsky. We prove that $V_{(G, \\theta)}$ is co$\\mathcal{CF}$ for any homomorphism $\\theta$ and finite group G by constructing a pushdown automaton and showing that the co-word problem of $V_{(G, \\theta)}$ is the cyclic shift of the language accepted by our automaton. A version of a conjecture due to Lehnert says that a group has context-free co-word problem exactly if it is a finitely generated subgroup of V. The groups $V_{(G,\\theta)}$ where $\\theta$ is not the identity homomorphism do not appear to have obvious embeddings into V, and may therefore be considered possible counterexamples to the conjecture. Demonstrative subgroups of $V$, which were introduced by Bleak and Salazar-Diaz, can be used to construct embeddings of certain wreath products and amalgamated free products into $V$. We extend the class of known finitely generated demonstrative subgroups of V to include all virtually cyclic groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-29T14:53:12.000Z",
      "abstract": "We describe a generalized Thompson group $V_{(G, \\theta)}$ for each finite group $G$ with homomorphism $\\theta$: $G \\rightarrow G$. Our group is constructed using the \\emph{cloning map} concept introduced by Witzel and Zaremsky. We prove that $V_{(G, \\theta)}$ is co$\\mathcal{CF}$ for any homomorphism $\\theta$ and finite group $G$ by constructing a pushdown automaton and showing that the co-word problem of $V_{(G, \\theta)}$ is the cyclic shift of the language our automaton accepts. Lehnert conjectures that a group has context-free co-word problem exactly if it is a finitely generated subgroup of $V$. The groups $V_{(G,\\theta)}$ where $\\theta$ is not the identity homomorphism do not appear to have obvious embeddings into $V$. We build off the work of Bleak and Salazar-Diaz regarding demonstrative groups. We extend the class of known finitely generated demonstrative subgroups of $V$ to include virtually cyclic groups. Additionally we construct embeddings for amalgamated free products where the amalgamated group is a demonstrative subgroup of $V$.",
      "comment": "14 pages, 9 figures",
      "journal": null,
      "doi": null,
      "authors": [
        "Rose Berns-Zieve",
        "Dana Fry",
        "Johnny Gillings",
        "Heather Mathews"
      ]
    },
    {
      "version": "v2",
      "updated": "2014-12-03T20:30:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "thompsons group",
      "embeddings",
      "finite group",
      "demonstrative subgroup",
      "homomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.7745B"
    }
  }
}