{
  "id": "1902.04883",
  "version": "v1",
  "published": "2019-02-13T13:13:18.000Z",
  "updated": "2019-02-13T13:13:18.000Z",
  "title": "A cubical flat torus theorem and some of its applications",
  "authors": [
    "Anthony Genevois"
  ],
  "comment": "34 pages, 10 figures. Comments are welcome!",
  "categories": [
    "math.GR"
  ],
  "abstract": "The article is dedicated to the proof of the following cubical version of the flat torus theorem. Let $G$ be a group acting on a CAT(0) cube complex $X$ and $A \\leq G$ a normal finitely generated abelian subgroup. Then there exists a median subalgebra $Y \\subset X$ which is $G$-invariant and which decomposes as a product of median algebras $T \\times F \\times Q$ such that: (1) the action $G \\curvearrowright Y$ decomposes as a product of actions $G \\curvearrowright T,F,Q$; (2) $F$ is a \\emph{flat}; (3) $Q$ is a finite-dimensional cube; (4) $A$ acts trivially on $T$. Some applications are included. For instance, a splitting theorem is proved and we show that a polycyclic group acting properly on a CAT(0) cube complex must be virtually abelian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-13T13:13:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "20F16"
    ],
    "keywords": [
      "cubical flat torus theorem",
      "applications",
      "cube complex",
      "normal finitely generated abelian subgroup",
      "median subalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}