{
  "id": "1808.00802",
  "version": "v1",
  "published": "2018-08-02T13:29:31.000Z",
  "updated": "2018-08-02T13:29:31.000Z",
  "title": "On Growth of Double Cosets in Hyperbolic Groups",
  "authors": [
    "Rita Gitik",
    "Eliyahu Rips"
  ],
  "comment": "two figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $gr(H,A,B)$ be the growth function of the double cosets $AhB, h \\in H$. We prove that the behavior of $gr(H,A,B)$ splits into two different cases. If $A$ and $B$ are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as $gr(H,A,B)$. We can even take $A=B$. In contrast, for quasiconvex subgroups A and B of infinite index, $gr(H,A,B)$ is exponential. Moreover, there exists a constant $\\lambda > 0$, such that $gr(H,A,B) >\\lambda f_H(r)$ for all big enough $r$, where $f_H(r)$ is the growth function of the group $H$. So, we have a clear dychotomy between the quasiconvex and non-quasiconvex case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-02T13:29:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F67",
      "20F65",
      "20B07"
    ],
    "keywords": [
      "hyperbolic group",
      "double cosets",
      "growth function",
      "non-quasiconvex case",
      "clear dychotomy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}