{
  "id": "2002.01388",
  "version": "v1",
  "published": "2020-02-04T16:14:47.000Z",
  "updated": "2020-02-04T16:14:47.000Z",
  "title": "Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups",
  "authors": [
    "Anthony Genevois",
    "Camille Horbez"
  ],
  "comment": "26 pages. Comments are welcome!",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We prove that the automorphism group of every infinitely-ended finitely generated group is acylindrically hyperbolic. In particular $\\mathrm{Aut}(\\mathbb{F}_n)$ is acylindrically hyperbolic for every $n\\ge 2$. More generally, if $G$ is a group which is not virtually cyclic, and hyperbolic relative to a finite collection $\\mathcal{P}$ of finitely generated proper subgroups, then $\\mathrm{Aut}(G,\\mathcal{P})$ is acylindrically hyperbolic. As a consequence, a free-by-cyclic group $\\mathbb{F}_n\\rtimes_{\\varphi}\\mathbb{Z}$ is acylindrically hyperbolic if and only if $\\varphi$ has infinite order in $\\mathrm{Out}(\\mathbb{F}_n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-04T16:14:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20E36",
      "20E06",
      "20F67",
      "20E08"
    ],
    "keywords": [
      "automorphism group",
      "infinitely-ended groups",
      "acylindrically hyperbolic",
      "acylindrical hyperbolicity",
      "finite collection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}