{
  "id": "1111.7048",
  "version": "v5",
  "published": "2011-11-30T05:04:58.000Z",
  "updated": "2014-12-02T06:37:33.000Z",
  "title": "Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces",
  "authors": [
    "F. Dahmani",
    "V. Guirardel",
    "D. Osin"
  ],
  "comment": "Revision, corrections and improvement of the exposition",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on $CAT(0)$ spaces, fundamental groups of graphs of groups, etc. We obtain a number of general results about rotating families and hyperbolically embedded subgroups; although our technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, we solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-07-26T09:24:02.000Z",
      "abstract": "We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of peripheral structures of relative hyperbolicity groups, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on CAT(0) spaces, fundamental groups of graphs of groups, etc. Although our technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, we solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.",
      "comment": "Revision, corrections and improvement of the exposition. Update of the problem section, ; 156 pages, many figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2014-12-02T06:37:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "20F06",
      "20E08",
      "57M27"
    ],
    "keywords": [
      "hyperbolically embedded subgroups",
      "hyperbolic spaces",
      "groups acting",
      "rotating families",
      "relatively hyperbolic groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 156,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.7048D"
    }
  }
}