{
  "id": "1009.1349",
  "version": "v2",
  "published": "2010-09-07T17:34:52.000Z",
  "updated": "2012-06-07T03:04:26.000Z",
  "title": "A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties",
  "authors": [
    "Meital Eliyahu",
    "David Garber",
    "Mina Teicher"
  ],
  "comment": "17 pages, 9 figures; final version, which corrects a mistake in the published version",
  "categories": [
    "math.GT",
    "math.AG",
    "math.GR"
  ],
  "abstract": "A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators $x_1, ..., x_n$ and the cyclic relations: $x_{i_k}x_{i_{k-1}} ... x_{i_1} = x_{i_{k-1}} ... x_{i_1} x_{i_k} = ... = x_{i_1} x_{i_k} ... x_{i_2}$ with no conjugations on the generators. We have already proved that if the graph of the arrangement is a disjoint union of cycles, then its fundamental group has a conjugation-free geometric presentation. In this paper, we extend this property to arrangements whose graphs are a disjoint union of cycle-tree graphs. Moreover, we study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We show that these presentations satisfy a completeness property in the sense of Dehornoy, if the corresponding graph of the arrangement has no edges. The completeness property is a powerful property which leads to many nice properties concerning the presentation (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-06-07T03:04:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H30",
      "32S22",
      "57M05",
      "20M05",
      "20F05"
    ],
    "keywords": [
      "conjugation-free geometric presentation",
      "fundamental group",
      "disjoint union",
      "completeness property",
      "line arrangements complement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.1349E"
    }
  }
}