{
  "id": "1905.03062",
  "version": "v1",
  "published": "2019-05-08T13:23:14.000Z",
  "updated": "2019-05-08T13:23:14.000Z",
  "title": "The geometry of groups containing almost normal subgroups",
  "authors": [
    "Alexander Margolis"
  ],
  "comment": "48 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "A subgroup $H\\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse $PD_n$ subgroup $H$ with $e(G/H)=\\infty$, then whenever $G'$ is quasi-isometric to $G$, it contains an almost normal subgroup $H'$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G/H$ and $G'/H'$ are quasi-isometric. This generalises a theorem of Mosher-Sageev-Whyte, who prove the case in which $G/H$ is quasi-isometric to a finite valence bushy tree. Using work of Mosher, we generalise a result of Farb-Mosher to show that for many surface group extensions $\\Gamma_L$, any group quasi-isometric to $\\Gamma_L$ is virtually isomorphic to $\\Gamma_L$. We also prove quasi-isometric rigidity for the class of finitely presented $\\mathbb{Z}$-by-($\\infty$ ended) groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-08T13:23:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20E08",
      "20J05",
      "57M07"
    ],
    "keywords": [
      "normal subgroup",
      "groups containing",
      "finite valence bushy tree",
      "surface group extensions",
      "well-defined quotient space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}