{
  "id": "0911.3385",
  "version": "v3",
  "published": "2009-11-17T19:41:10.000Z",
  "updated": "2010-06-26T01:59:01.000Z",
  "title": "A relationship between twisted conjugacy classes and the geometric invariants $Ω^n$",
  "authors": [
    "Nic Koban",
    "Peter Wong"
  ],
  "comment": "13 pages",
  "journal": "Geom. Dedicata (2011) 151:233-243",
  "categories": [
    "math.GR",
    "math.AT"
  ],
  "abstract": "A group $G$ is said to have the property $R_\\infty$ if every automorphism $\\phi \\in {\\rm Aut}(G)$ has an infinite number of $\\phi$-twisted conjugacy classes. Recent work of Gon\\c{c}alves and Kochloukova uses the $\\Sigma^n$ (Bieri-Neumann-Strebel-Renz) invariants to show the $R_{\\infty}$ property for a certain class of groups, including the generalized Thompson's groups $F_{n,0}$. In this paper, we make use of the $\\Omega^n$ invariants, analogous to $\\Sigma^n$, to show $R_{\\infty}$ for certain finitely generated groups. In particular, we give an alternate and simpler proof of the $R_{\\infty}$ property for BS(1,n). Moreover, we give examples for which the $\\Omega^n$ invariants can be used to determine the $R_{\\infty}$ property while the $\\Sigma^n$ invariants techniques cannot.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-06-26T01:59:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "E45",
      "55M20",
      "57M07"
    ],
    "keywords": [
      "twisted conjugacy classes",
      "geometric invariants",
      "relationship",
      "infinite number",
      "generalized thompsons groups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.3385K"
    }
  }
}