{
  "id": "1405.0994",
  "version": "v2",
  "published": "2014-05-05T19:06:58.000Z",
  "updated": "2014-11-02T23:47:32.000Z",
  "title": "Residual nilpotence and ordering in one-relator groups and knot groups",
  "authors": [
    "I. M. Chiswell",
    "A. M. W. Glass",
    "John S. Wilson"
  ],
  "comment": "Minor changes, references added",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G=< x,t\\mid w>$ be a one-relator group, where $w$ is a word in $x,t$. If $w$ is a product of conjugates of $x$ then, associated with $w$, there is a polynomial $A_w(X)$ over the integers, which in the case when $G$ is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on $w$, that if all roots of $A_w(X)$ are real and positive then $G$ is bi-orderable, and that if $G$ is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-05T19:06:58.000Z",
      "abstract": "Let $G=\\langle x,t\\mid w\\rangle$ be a one-relator group, where $w$ is a word in $x,t$. If $w$ is a product of conjugates of $x$ then, associated with $w$, there is a polynomial $A_w(X)$ over the integers, which in the case when $G$ is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on $w$, that if all roots of $A_w(X)$ are real and positive then $G$ is bi-orderable, and that if $G$ is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-02T23:47:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "knot group",
      "one-relator group",
      "residual nilpotence",
      "results relies",
      "sheds light"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1017/S0305004114000644",
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2015,
      "month": "Mar",
      "volume": 158,
      "number": 2,
      "pages": 275
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015MPCPS.158..275C"
    }
  }
}