{
  "id": "1605.09067",
  "version": "v1",
  "published": "2016-05-29T21:33:28.000Z",
  "updated": "2016-05-29T21:33:28.000Z",
  "title": "Alexander and Thurston norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groups",
  "authors": [
    "Florian Funke",
    "Dawid Kielak"
  ],
  "comment": "38 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We investigate Friedl-L\\\"uck's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\\mathbb{Z}$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. For descending HNN extensions of $F_2$, we prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the universal $L^2$-torsion of a descending HNN extension of $F_2$ locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. When the HNN extension is taken over $F_n$ along a polynomially growing automorphism with unipotent image in $GL(n, \\mathbb{Z})$, we show that the Newton polytope of the universal $L^2$-torsion and the BNS invariant completely determine one another.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-29T21:33:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20E06",
      "16S85"
    ],
    "keywords": [
      "bieri-neumann-strebel invariant",
      "thurston norm",
      "descending hnn extension",
      "free-by-cyclic groups",
      "newton polytope"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}