{
  "id": "1709.06524",
  "version": "v1",
  "published": "2017-09-19T16:44:03.000Z",
  "updated": "2017-09-19T16:44:03.000Z",
  "title": "Almost-automorphisms of trees, cloning systems and finiteness properties",
  "authors": [
    "Rachel Skipper",
    "Matthew C. B. Zaremsky"
  ],
  "comment": "44 pages, 11 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "We prove that the group of almost-automorphisms of the infinite rooted regular $d$-ary tree $\\mathcal{T}_d$ arises naturally as the Thompson-like group of a so called $d$-ary cloning system. A similar phenomenon occurs for any Nekrashevych group $V_d(G)$, for $G\\le Aut(\\mathcal{T}_d)$ a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the R\\\"over group, using the Grigorchuk group for $G$, is of type $F_\\infty$. Namely, we find some natural conditions on subgroups of $G$ to ensure that $V_d(G)$ is of type $F_\\infty$, and in particular we prove this for all $G$ in the infinite family of \\v{S}uni\\'c groups. We also prove that if $G$ is itself of type $F_\\infty$ then so is $V_d(G)$, and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group $G$ yields a type $F_\\infty$ Nekrashevych group $V_d(G)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-19T16:44:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "57M07"
    ],
    "keywords": [
      "cloning system",
      "finiteness properties",
      "finitely generated virtually free group",
      "almost-automorphisms",
      "nekrashevych group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}