{
  "id": "2008.07639",
  "version": "v1",
  "published": "2020-08-17T21:45:31.000Z",
  "updated": "2020-08-17T21:45:31.000Z",
  "title": "Planar boundaries and parabolic subgroups",
  "authors": [
    "G. Christopher Hruska",
    "Genevieve S. Walsh"
  ],
  "comment": "25 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We study the Bowditch boundaries of relatively hyperbolic group pairs, focusing on the case where there are no cut points. We show that, if there are parabolic inseparable cut pairs in $\\partial(G,\\mathcal{P})$, the group $G$ splits over a finite group. We use this to prove that when $G$ is one ended, and $(G, \\mathcal{P})$ is a relatively hyperbolic pair with connected planar boundary with no cut points, then every element of $\\mathcal{P}$ is virtually a surface group. This conclusion is consistent with the conjecture that such a group $G$ is virtually Kleinian. We give numerous examples to show the necessity of our assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-17T21:45:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F67",
      "20E08"
    ],
    "keywords": [
      "parabolic subgroups",
      "cut points",
      "relatively hyperbolic group pairs",
      "parabolic inseparable cut pairs",
      "connected planar boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}