{
  "id": "0711.3806",
  "version": "v4",
  "published": "2007-11-24T01:43:24.000Z",
  "updated": "2009-02-21T19:47:37.000Z",
  "title": "Geometric Intersection Number and analogues of the Curve Complex for free groups",
  "authors": [
    "Ilya Kapovich",
    "Martin Lustig"
  ],
  "comment": "Revised version, to appear in Geometry & Topology",
  "journal": "Geometry and Topology, vol. 13 (2009), no. 3, pp. 1805-1833",
  "doi": "10.2140/gt.2009.13.1805",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "For the free group $F_{N}$ of finite rank $N \\geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \\emph{geometric intersection form} \\[ <, >: \\bar{cv}(F_N)\\times Curr(F_N)\\to \\mathbb R_{\\ge 0}. \\] Here $\\bar{cv}(F_N)$ is the closure of unprojectivized Culler-Vogtmann's Outer space $cv(F_N)$ in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $\\bar{cv}(F_N)$ consists of all \\emph{very small} minimal isometric actions of $F_N$ on $\\mathbb R$-trees. The projectivization of $\\bar{cv}(F_N)$ provides a free group analogue of Thurston's compactification of the Teichm\\\"uller space. As an application, using the \\emph{intersection graph} determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-02-21T19:47:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geometric intersection number",
      "curve complex",
      "intersection form",
      "equivariant gromov-hausdorff convergence topology",
      "unprojectivized culler-vogtmanns outer space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.3806K"
    }
  }
}