{
  "id": "2009.13314",
  "version": "v1",
  "published": "2020-09-28T13:37:08.000Z",
  "updated": "2020-09-28T13:37:08.000Z",
  "title": "Thermodynamic metrics on outer space",
  "authors": [
    "Tarik Aougab",
    "Matt Clay",
    "Yo'av Rieck"
  ],
  "comment": "62 pages, 7 figures",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "In this paper we consider two piecewise Riemannian metrics defined on the Culler-Vogtmann outer space which we call the entropy metric and the pressure metric. As a result of work of McMullen, these metrics can be seen as analogs of the Weil-Petersson metric on the Teichm\\\"uller space of a closed surface. We show that while the geometric analysis of these metrics is similar to that of the Weil-Petersson metric, from the point of view of geometric group theory, these metrics behave very differently to the Weil-Petersson metric. Specifically, we show that when the rank $r$ is at least 4, the action of ${\\rm Out}(\\mathbb{F}_r)$ on the completion of the Culler-Vogtmann outer space using the entropy metric has a fixed point. A similar statement also holds for the pressure metric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T13:37:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E05",
      "20F65"
    ],
    "keywords": [
      "thermodynamic metrics",
      "weil-petersson metric",
      "culler-vogtmann outer space",
      "entropy metric",
      "pressure metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}