{
  "id": "1509.04862",
  "version": "v1",
  "published": "2015-09-16T09:37:40.000Z",
  "updated": "2015-09-16T09:37:40.000Z",
  "title": "An application of the Local C(G,T) Theorem to a conjecture of Weiss",
  "authors": [
    "Pablo Spiga"
  ],
  "comment": "to appear on the Bulletin of the London Math. Society",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Let $\\Gamma$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex of $\\Gamma$ and let $G_v^{\\Gamma(v)}$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\\Gamma(v)$. The graph $\\Gamma$ is said to be $G$-\\emph{locally primitive} if $G_v^{\\Gamma(v)}$ is primitive. Richard Weiss conjectured in $1978$ that, there exists a function $f:\\mathbb{N}\\to \\mathbb{N}$ such that, if $\\Gamma$ is a connected $G$-vertex-transitive locally primitive graph of valency $d$ and $v$ is a vertex of $\\Gamma$ with $|G_v|$ finite, then $|G_v|\\leq f(d)$. As an application of the Local $C(G,T)$ Theorem, we prove this conjecture when $G_v^{\\Gamma(v)}$ contains an abelian regular subgroup. In fact, we show that the point-wise stabiliser in $G$ of a ball of $\\Gamma$ of radius $4$ is the identity subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-16T09:37:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "application",
      "conjecture",
      "abelian regular subgroup",
      "permutation group",
      "identity subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150904862S"
    }
  }
}