{
  "id": "1111.5589",
  "version": "v1",
  "published": "2011-11-23T19:20:46.000Z",
  "updated": "2011-11-23T19:20:46.000Z",
  "title": "Deformations and rigidity of lattices in solvable Lie groups",
  "authors": [
    "Oliver Baues",
    "Benjamin Klopsch"
  ],
  "doi": "10.1112/jtopol/jtt016",
  "categories": [
    "math.DG",
    "math.GR"
  ],
  "abstract": "Let $G$ be a simply connected, solvable Lie group and $\\Gamma$ a lattice in $G$. The deformation space $\\mathcal{D}(\\Gamma,G)$ is the orbit space associated to the action of $\\Aut(G)$ on the space $\\mathcal{X}(\\Gamma,G)$ of all lattice embeddings of $\\Gamma$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev and Sait\\^o for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice $\\Gamma$ in $G$ is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of $G$ is connected. This implies that every lattice in a solvable Lie group virtually embeds as a Zariski-dense lattice with finite deformation space. We give examples of solvable Lie groups $G$ which admit Zariski-dense lattices $\\Gamma$ such that $\\mathcal{D}(\\Gamma,G)$ is countably infinite, and also examples where the maximal nilpotent normal subgroup of $G$ is connected and simultaneously $G$ has lattices with uncountable deformation space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-23T19:20:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E40",
      "22E25",
      "20F16"
    ],
    "keywords": [
      "maximal nilpotent normal subgroup",
      "deformation space",
      "zariski-dense lattice",
      "solvable lie group virtually embeds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.5589B"
    }
  }
}