{
  "id": "1905.00738",
  "version": "v1",
  "published": "2019-05-01T07:41:15.000Z",
  "updated": "2019-05-01T07:41:15.000Z",
  "title": "A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube complexes",
  "authors": [
    "Anthony Genevois"
  ],
  "comment": "15 pages, 3 figures. Comments are welcome. arXiv admin note: text overlap with arXiv:1902.04883",
  "categories": [
    "math.GR"
  ],
  "abstract": "In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group $F$, the lamplighter group $F \\wr \\mathbb{F}_2$ over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on a infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, roughly speaking, given a group $G$ acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup $L \\leq G$, either the normaliser $N_G(L)$ is close to be free abelian or, for every $k \\geq 1$, $N_G(L)$ contains a non-abelian free subgroup commuting with a subgroup of $L$ of size $\\geq k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-01T07:41:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67"
    ],
    "keywords": [
      "cube complex",
      "groups acting",
      "general criterion",
      "non-trivial finite group",
      "infinite torsion subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}