{
  "id": "1008.0417",
  "version": "v3",
  "published": "2010-08-02T21:53:22.000Z",
  "updated": "2019-04-16T14:27:19.000Z",
  "title": "Discriminantal bundles, arrangement groups, and subdirect products of free groups",
  "authors": [
    "Daniel C. Cohen",
    "Michael J. Falk",
    "Richard C. Randell"
  ],
  "comment": "33 pages, 7 figures. This version is a substantial revision, with more general formulations and corrected proofs of the results of Section 3, and a new observation about decomposable arrangements",
  "categories": [
    "math.GT",
    "math.CO",
    "math.GR"
  ],
  "abstract": "A generating set for a complex hyperplane arrangement A is a collection of continuous functions on the complement M of A whose differences are non-vanishing on M. A generating set F gives rise to a space of extensions E_k(A,F) of A which is the total space of a bundle over M. One can use such bundles to construct representations of the fundamental group of M analogous to the Lawrence-Krammer-Bigelow representation of the pure braid group. We construct generating sets for several families of arrangements. In this context we analyze the kernel of the homomorphism p_X from an arbitrary group G with finite generating set Y to the product of quotients G_S of G corresponding to an antichain X of 2^Y. Generalizing an argument of T. Stanford concerning Brunnian braids, we show the kernel of p_X is generated by iterated commutators that are transverse to X in a certain sense, provided the projections of G to G_S split compatibly. Under this hypothesis, we derive a test for the injectivity of p_X that depends only on 2- and 3-fold commutators of generators. We show that p_X is injective for some well-studied decomposable arrangements. If A is central and X consists of rank two flats, the homomorphism p_X induces a homomorphism from the quotient of G by Z to a product of free groups. We show this map is injective if and only if p_X is, provided the incidence graph L of X with A is connected. The image is a normal subgroup, and the cokernel is equal to H^1(L,Z). Using results of Meier-Meinert-Van Wyk on subdirect products of free groups, we then identify the cohomological finiteness type of the image in terms of L, and deduce the cohomological finiteness type of several arrangement groups. Finally we show that a decomposable arrangement group either has a conjugation-free presentation or is not residually nilpotent.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-08-04T16:41:06.000Z",
      "abstract": "The Lawrence-Krammer-Bigelow representation of the braid group arises from the monodromy representation on the twisted homology of the fiber of a certain fiber bundle in which the base and total space are complements of braid arrangements, and the fiber is the complement of a discriminantal arrangement. We present a more general version of this construction and use it to construct nontrivial bundles on the complement of an arbitrary arrangement \\A\\ whose fibers are products of discriminantal arrangements. This leads us to consider the natural homomorphism $\\rho_\\X$ from the arrangement group $G(\\A)=\\pi_1(\\C^\\ell - \\bigcup \\A)$ to the product of groups $G(\\A_X), X \\in \\X,$ corresponding to a set \\X\\ of rank-two flats. Generalizing an argument of T. Stanford, we describe the kernel in terms of iterated commutators, when generators of $G(\\A_X), X \\in \\X,$ can be chosen compatibly. We use this to derive a test for injectivity of $\\rho_\\X.$ We show $\\rho_\\X$ is injective for several well-studied decomposable arrangements. If \\A\\ is central, the homomorphism $\\rho_\\X$ induces a natural homomorphism $\\bar{\\rho}_\\X$ from the projectivized group $\\bar{G}(\\A)$ into the product $\\prod_{X\\in \\X} \\bar{G}(\\A_X),$ whose factors are free groups. We show $\\bar{\\rho}_\\X$ is injective if and only if $\\rho_\\X$ is. In this case $\\bar{G}$ is isomorphic to a specific finitely-presented, combinatorially-determined subdirect product of free groups. In particular $\\bar{G}$ is residually free, residually torsionfree nilpotent, a-T-menable, and linear. We show the image of $\\bar{\\rho}_\\X$ is a normal subgroup with free abelian quotient, and compute the rank of the quotient in terms of the incidence graph of \\X\\ with \\A. When $\\rho_\\X$ is injective, we conclude $\\bar{G}$ is of type $F_{m-1}$ and not of type $F_m,$ $m=|\\X|.$",
      "comment": "29 pages, 9 figures",
      "journal": null,
      "doi": null,
      "authors": [
        "Daniel C. Cohen",
        "Michael Falk",
        "Richard Randell"
      ]
    },
    {
      "version": "v3",
      "updated": "2019-04-16T14:27:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "32S22",
      "52C35",
      "55R80"
    ],
    "keywords": [
      "free groups",
      "arrangement group",
      "subdirect product",
      "discriminantal bundles",
      "discriminantal arrangement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.0417C"
    }
  }
}