{
  "id": "math/0511647",
  "version": "v3",
  "published": "2005-11-27T00:23:26.000Z",
  "updated": "2006-07-07T18:45:36.000Z",
  "title": "Quasi-isometries and rigidity of solvable groups",
  "authors": [
    "Alex Eskin",
    "David Fisher",
    "Kevin Whyte"
  ],
  "comment": "19 pages, 5 figures. One theorem added concerning quasi-isometric rigidity of three manifold groups",
  "categories": [
    "math.GR",
    "math.GT",
    "math.MG"
  ],
  "abstract": "In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \\ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \\ltimes \\R^n$ proves a conjecture made by Farb and Mosher in [FM4]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as \"coarse differentiation\".",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-07-07T18:45:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasi-isometry",
      "finitely generated group",
      "solvable groups",
      "groups quasi-isometric",
      "dimenionsional solvable lie group sol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11647E"
    }
  }
}