{
  "id": "1708.02855",
  "version": "v1",
  "published": "2017-08-09T14:31:29.000Z",
  "updated": "2017-08-09T14:31:29.000Z",
  "title": "Local cut points and splittings of relatively hyperbolic groups",
  "authors": [
    "Matthew Haulmark"
  ],
  "comment": "35 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "In this paper we show that the existence of a non-parabolic local cut point in the Bowditch boundary $\\partial(G,\\mathbb{P})$ of a relatively hyperbolic group $(G,\\mathbb{P})$ implies that $G$ splits over a $2$-ended subgroup. This theorem generalizes a theorem of Bowditch from the setting of hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of $1$-dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over $2$-ended subgroups and no peripheral splittings. In order to prove the boundary classification result we require a notion of ends of a group which is more general than the standard notion. We show that if a finitely generated discrete group acts properly and cocompactly on two generalized Peano continua $X$ and $Y$, then $Ends(X)$ is homeomorphic to $Ends(Y)$. Thus we propose an alternate definition of $Ends(G)$ which increases the class of spaces on which $G$ can act.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-09T14:31:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67"
    ],
    "keywords": [
      "relatively hyperbolic group",
      "generated discrete group acts",
      "splittings",
      "non-parabolic local cut point",
      "bowditch boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}