{
  "id": "1812.09035",
  "version": "v1",
  "published": "2018-12-21T10:23:06.000Z",
  "updated": "2018-12-21T10:23:06.000Z",
  "title": "Commensurators of abelian subgroups in CAT(0) groups",
  "authors": [
    "Jingyin Huang",
    "Tomasz Prytuła"
  ],
  "comment": "20 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We study the structure of the commensurator of a virtually abelian subgroup $H$ in $G$, where $G$ acts properly on a $\\mathrm{CAT}(0)$ space $X$. When $X$ is a Hadamard manifold and $H$ is semisimple, we show that the commensurator of $H$ coincides with the normalizer of a finite index subgroup of $H$. When $X$ is a $\\mathrm{CAT}(0)$ cube complex or a thick Euclidean building and the action of $G$ is cellular, we show that the commensurator of $H$ is an ascending union of normalizers of finite index subgroups of $H$. We explore several special cases where the results can be strengthened and we discuss a few examples showing the necessity of various assumptions. Finally, we present some applications to the constructions of classifying spaces with virtually abelian stabilizers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-21T10:23:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67"
    ],
    "keywords": [
      "commensurator",
      "finite index subgroup",
      "virtually abelian subgroup",
      "hadamard manifold",
      "normalizer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}