{
  "id": "1908.07664",
  "version": "v1",
  "published": "2019-08-21T01:16:42.000Z",
  "updated": "2019-08-21T01:16:42.000Z",
  "title": "On graphic arrangement groups",
  "authors": [
    "Daniel C Cohen",
    "Michael J Falk"
  ],
  "comment": "25 pages, 1 figure",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "A finite simple graph $\\Gamma$ determines a quotient $P_\\Gamma$ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work with R. Randell. We show that, for a $K_4$-free graph $\\Gamma$, a product of deletion maps is injective, embedding $P_\\Gamma$ in a product of free groups. Then $P_\\Gamma$ is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show $P_\\Gamma$ is of homological finiteness type $F_{m-1}$, but not $F_m$, where $m$ is the number of copies of $K_3$ in $\\Gamma$, except in trivial cases. The embedding result is extended to graphs whose 4-cliques share at most one edge, giving an injection of $P_\\Gamma$ into the product of pure braid groups corresponding to maximal cliques of $\\Gamma$. We give examples showing that this map may inject in more general circumstances. We define the graphic braid group $B_\\Gamma$ as a natural extension of $P_\\Gamma$ by the automorphism group of $\\Gamma$, and extend our homological finiteness result to these groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-21T01:16:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "32S22",
      "52C35",
      "20E26"
    ],
    "keywords": [
      "graphic arrangement group",
      "pure braid group",
      "graphic braid group",
      "finite simple graph",
      "prior joint work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}