{
  "id": "1312.3374",
  "version": "v1",
  "published": "2013-12-12T00:26:23.000Z",
  "updated": "2013-12-12T00:26:23.000Z",
  "title": "Controlled Connectivity for Semi-Direct Products Acting on Locally Finite Trees",
  "authors": [
    "Keith Jones"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "In 2003 Bieri and Geoghegan generalized the Bieri-Neuman-Strebel invariant $\\Sigma^1$ by defining $\\Sigma^1(\\rho)$, $\\rho$ an isometric action by a finitely generated group $G$ on a proper CAT(0) space $M$. In this paper, we show how the natural and well-known connection between Bass-Serre theory and covering space theory provides a framework for the calculation of $\\Sigma^1(\\rho)$ when $\\rho$ is a cocompact action by $G = B \\rtimes A$, $A$ a finitely generated group, on a locally finite Bass-Serre tree $T$ for $A$. This framework leads to a theorem providing conditions for including an endpoint in, or excluding an endpoint from, $\\Sigma^1(\\rho)$. When $A$ is a finitely generated free group acting on its Cayley graph, we can restate this theorem from a more algebraic perspective, which leads to some general results on $\\Sigma^1$ for such actions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-12T00:26:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locally finite trees",
      "semi-direct products acting",
      "controlled connectivity",
      "generated free group acting",
      "finitely generated group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.3374J"
    }
  }
}