{
  "id": "1111.2499",
  "version": "v3",
  "published": "2011-11-10T15:21:30.000Z",
  "updated": "2018-09-14T10:31:31.000Z",
  "title": "Quasi-hyperbolic planes in relatively hyperbolic groups",
  "authors": [
    "John M. Mackay",
    "Alessandro Sisto"
  ],
  "comment": "v1: 32 pages, 4 figures. v2: 38 pages, 4 figures. v3: 44 pages, 4 figures. An application (Theorem 1.2) is weakened as there was an error in its proof in section 7, all other changes minor, improved exposition",
  "categories": [
    "math.GR",
    "math.GT",
    "math.MG"
  ],
  "abstract": "We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to \"almost every\" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connectedness. In particular, we prove a new existence result for quasi-arcs that avoid obstacles.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-12T15:27:24.000Z",
      "abstract": "We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit certain splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. The specific embeddings we find remain quasi-isometric embeddings when composed with the natural map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to \"almost every\" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The proofs of these theorems involve quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connectedness. In particular, we prove a new existence result for quasi-arcs that avoid obstacles.",
      "comment": "v1: 32 pages, 4 figures. v2: 38 pages, 4 figures. Improved exposition and minor corrections",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2018-09-14T10:31:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "51F99"
    ],
    "keywords": [
      "relatively hyperbolic groups",
      "quasi-hyperbolic planes",
      "study quantitative geometric properties",
      "admit peripheral splittings",
      "remain quasi-isometric embeddings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.2499M"
    }
  }
}