{
  "id": "2010.08035",
  "version": "v1",
  "published": "2020-10-15T21:46:13.000Z",
  "updated": "2020-10-15T21:46:13.000Z",
  "title": "Finiteness Properties of Locally Defined Groups",
  "authors": [
    "Daniel S. Farley",
    "Bruce Hughes"
  ],
  "comment": "61 pages, no figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $X$ be a set and let $S$ be an inverse semigroup of partial bijections of $X$. Thus, an element of $S$ is a bijection between two subsets of $X$, and the set $S$ is required to be closed under the operations of taking inverses and compositions of functions. We define $\\Gamma_{S}$ to be the set of self-bijections of $X$ in which each $\\gamma \\in \\Gamma_{S}$ is expressible as a union of finitely many members of $S$. This set is a group with respect to composition. The groups $\\Gamma_{S}$ form a class containing numerous widely studied groups, such as Thompson's group $V$, the Nekrashevych-R\\\"{o}ver groups, Houghton's groups, and the Brin-Thompson groups $nV$, among many others. We offer a unified construction of geometric models for $\\Gamma_{S}$ and a general framework for studying the finiteness properties of these groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-15T21:46:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20J05",
      "20M18"
    ],
    "keywords": [
      "locally defined groups",
      "finiteness properties",
      "inverse semigroup",
      "composition",
      "partial bijections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}