{
  "id": "math/0507301",
  "version": "v1",
  "published": "2005-07-14T20:27:11.000Z",
  "updated": "2005-07-14T20:27:11.000Z",
  "title": "The Large Scale Geometry of Nilpotent-by-Cyclic Groups",
  "authors": [
    "Ashley Reiter Ahlin"
  ],
  "comment": "Ph.D. dissertation at University of Chicago Mathematics Department, June 2002",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces an injective endomorphism phi of the Lie algebra g associated to the Lie group in which N is a lattice. A normal form for automorphisms of nilpotent Lie algebras--permuted absolute Jordan form-- is defined and conjectured to be a quasi-isometry invariant. We show that if phi, theta are endomorphisms of lattices in a fixed Carnot group G, and if the induced automorphisms of g have the same permuted absolute Jordan form, then Gamma_phi and Gamma_theta are quasi-isometric. Two quasi-isometry invariants are also found: the set of ``divergence rates'' of vertical flow lines: D_phi the ``growth spaces'': g_n subset g These do not establish that permuted absolute Jordan form is a quasi-isometry invariant, although they are major steps toward that conjecture. Furthermore, the quasi-isometric rigidity of finitely-presented nilpotent-by-cyclic groups is proven: any finitely-presented group quasi-isometric to a nonpolycyclic nilpotent-by-cyclic group is (virtually-nilpotent)-by-cyclic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-07-14T20:27:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large scale geometry",
      "permuted absolute jordan form",
      "lie algebras-permuted absolute jordan",
      "quasi-isometry invariant",
      "quasi-isometric"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7301R"
    }
  }
}