{
  "id": "1002.0996",
  "version": "v4",
  "published": "2010-02-04T13:43:10.000Z",
  "updated": "2017-04-20T19:44:12.000Z",
  "title": "Cannon-Thurston Maps for Kleinian Groups",
  "authors": [
    "Mahan Mj"
  ],
  "comment": "39 pgs 1 fig. Final version incorporating referee comments. To appear in Forum of Mathematics, Pi",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to end-points of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon-Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-06-07T12:23:38.000Z",
      "abstract": "We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups, proving a conjecture of McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to end-points of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon-Thurston maps for arbitrary finitely generated Kleinian groups.",
      "comment": "28 pages 1 figure. The first part of this paper was contained in an earlier version of arXiv:math/0607509. The second part of the paper describes point pre-images. v2: Revised to allow parabolics. Completes the project of proving the existence and describing the structure of Cannon-Thurston maps for arbitrary finitely generated Kleinian groups. v3: Many expository changes. Several details added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2017-04-20T19:44:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M50",
      "20F67",
      "20F65",
      "22E40"
    ],
    "keywords": [
      "cannon-thurston maps",
      "arbitrary finitely generated kleinian groups",
      "degenerate free groups",
      "point pre-images",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.0996M"
    }
  }
}