{
  "id": "1212.0216",
  "version": "v2",
  "published": "2012-12-02T15:09:34.000Z",
  "updated": "2013-12-08T12:06:49.000Z",
  "title": "Remarks on minimal sets and conjectures of Cassels, Swinnerton-Dyer, and Margulis",
  "authors": [
    "Jinpeng An",
    "Barak Weiss"
  ],
  "comment": "18 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove that a hypothesis of Cassels, Swinnerton-Dyer, recast by Margulis as statement on the action of the diagonal group $A$ on the space of unimodular lattices, is equivalent to several assertions about minimal sets for this action. More generally, for a maximal $\\mathbb{R}$-diagonalizable subgroup $A$ of a reductive group $G$ and a lattice $\\Gamma$ in $G$, we give a sufficient condition for a compact $A$-minimal subset $Y$ of $G/\\Gamma$ to be of a simple form, which is also necessary if $G$ is $\\mathbb{R}$-split. We also show that the stabilizer of $Y$ has no nontrivial connected unipotent subgroups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-08T12:06:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimal sets",
      "swinnerton-dyer",
      "conjectures",
      "nontrivial connected unipotent subgroups",
      "simple form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.0216A"
    }
  }
}