{
  "id": "1401.0228",
  "version": "v2",
  "published": "2013-12-31T22:51:32.000Z",
  "updated": "2014-05-13T17:20:21.000Z",
  "title": "E-polynomial of SL(2,C)-Character Varieties of Free groups",
  "authors": [
    "Samuel Cavazos",
    "Sean Lawton"
  ],
  "comment": "23 pages; v2 has a more streamlined exposition, and includes additional references; to appear in International Journal of Mathematics",
  "journal": "Internat. J. Math. 25 (2014), no. 6, 1450058 (27 pages)",
  "doi": "10.1142/S0129167X1450058X",
  "categories": [
    "math.AG",
    "math.GN",
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $\\mathsf{F}_r$ be a free group of rank $r$, $\\mathbb{F}_q$ a finite field of order q, and let $\\mathrm{SL}_n(\\mathbb{F}_q)$ act on $\\mathrm{Hom}(\\mathsf{F}_r, \\mathrm{SL}_n(\\mathbb{F}_q))$ by conjugation. We describe a general algorithm to determine the cardinality of the set of orbits $\\mathrm{Hom}(\\mathsf{F}_r, \\mathrm{SL}_n(\\mathbb{F}_q))/\\mathrm{SL}_n(\\mathbb{F}_q)$. Our first main theorem is the implementation of this algorithm in the case $n=2$. As an application, we determine the $E$-polynomial of the character variety $\\mathrm{Hom}(\\mathsf{F}_r, \\mathrm{SL}_2(\\mathbb{C}))//!/\\mathrm{SL}_2(\\mathbb{C})$, and of its smooth and singular locus. Thus we determine the Euler characteristic of these spaces.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-13T17:20:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "14D20",
      "14G05",
      "14G15"
    ],
    "keywords": [
      "free group",
      "e-polynomial",
      "first main theorem",
      "character variety",
      "euler characteristic"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.0228C"
    }
  }
}