{
  "id": "math/0404118",
  "version": "v1",
  "published": "2004-04-06T04:34:41.000Z",
  "updated": "2004-04-06T04:34:41.000Z",
  "title": "Elementary subgroups of relatively hyperbolic groups and bounded generation",
  "authors": [
    "D. V. Osin"
  ],
  "comment": "21 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a group hyperbolic relative to a collection of subgroups $\\{H_\\lambda ,\\lambda \\in \\Lambda \\} $. We say that a subgroup $Q\\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\\{H_\\lambda ,\\lambda \\in \\Lambda \\} \\cup \\{Q\\} $. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element $g\\in G$ has infinite order and is not conjugate to an element of $H_\\lambda $, $\\lambda \\in \\Lambda $, then the (unique) maximal elementary subgroup contained $g$ is hyperbolically embedded into $G$. This allows to prove that if $G$ is boundedly generated, then $G$ is elementary or $H_\\lambda =G$ for some $\\lambda \\in \\Lambda $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-04-06T04:34:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "20F69"
    ],
    "keywords": [
      "relatively hyperbolic groups",
      "bounded generation",
      "infinite order",
      "maximal elementary subgroup",
      "hyperbolically embedded subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4118O"
    }
  }
}