{
  "id": "1306.3403",
  "version": "v4",
  "published": "2013-06-14T14:11:01.000Z",
  "updated": "2016-10-30T21:43:58.000Z",
  "title": "Limit sets for modules over groups on CAT(0) spaces -- from the Euclidean to the hyperbolic",
  "authors": [
    "Robert Bieri",
    "Ross Geoghegan"
  ],
  "comment": "This is the final published version",
  "journal": "Proc. London Math. Soc. (2016) 112 (6): 1059-1102",
  "doi": "10.1112/plms/pdw018",
  "categories": [
    "math.GR"
  ],
  "abstract": "The observation that the 0-dimensional Geometric Invariant $\\Sigma ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincar\\'e's limit set $\\Lambda (\\Gamma)$ of a discrete group $\\Gamma $ of M\\\"obius transformations (which contains the horospherical limit set of $\\Gamma $) to the roots of tropical geometry (closely related to $\\Sigma ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $\\Sigma (M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is a subset of the boundary at infinity of M. On the way we meet instances where $\\Sigma (M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-17T15:29:15.000Z",
      "abstract": "The observation that the 0-dimensional Geometric Invariant $\\Sigma ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincar\\'{e}'s limit set $\\Lambda (\\Gamma)$ of a discrete group $\\Gamma $ of M\\\"{o}bius transformations (which contains the horospherical limit set of $\\Gamma $) to the roots of tropical geometry (closely related to $\\Sigma ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $\\Sigma (M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is a subset of the boundary at infinity of M. On the way we meet instances where $\\Sigma (M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2016-10-30T21:43:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20E42",
      "14T05"
    ],
    "keywords": [
      "hyperbolic",
      "horospherical limit set opens",
      "direct trail",
      "thurston norm",
      "discrete group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.3403B"
    }
  }
}