{
  "id": "1912.08610",
  "version": "v1",
  "published": "2019-12-17T17:35:40.000Z",
  "updated": "2019-12-17T17:35:40.000Z",
  "title": "Symmetrical 2-extensions of the 3-dimensional grid. I",
  "authors": [
    "Kirill Kostousov"
  ],
  "comment": "164 pages, 5 tables (2 of them are long)",
  "categories": [
    "math.CO",
    "math.AG",
    "math.GR"
  ],
  "abstract": "For a positive integer $d$, a connected graph $\\Gamma$ is a symmetrical 2-extension of the $d$-dimensional grid $\\Lambda^d$ if there exists a vertex-tran\\-sitive group $G$ of automorphisms of $\\Gamma$ and its imprimitivity system $\\sigma$ with blocks of order 2 such that there exists an isomorphism $\\varphi$ of the quotient graph $\\Gamma/\\sigma$ onto $\\Lambda^d$. The tuple $(\\Gamma, G, \\sigma, \\varphi)$ with specified components is called a realization of the symmetrical 2-extension $\\Gamma$ of the grid $\\Lambda^{d}$. Two realizations $(\\Gamma_1, G_1,$ $\\sigma_1, \\varphi_1)$ and $(\\Gamma_2, G_2, \\sigma_2, \\varphi_2)$ are called equivalent if there exists an isomorphism of the graph $\\Gamma_1$ onto $\\Gamma_2$ which maps $\\sigma_1$ onto $\\sigma_2$. V. Trofimov proved that, up to equivalence, there are only finitely many realizations of symmetrical $2$-extensions of $\\Lambda^{d}$ for each positive integer $d$. E. Konovalchik and K. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid $\\Lambda^2$. In this work we found all, up to equivalence, realizations $(\\Gamma, G, \\sigma, \\varphi)$ of symmetrical 2-extensions of the grid $\\Lambda^3$ for which only the trivial automorphism of $\\Gamma$ preserves all blocks of $\\sigma$ (we prove that there are 5573 such realizations, and that among corresponding graphs $\\Gamma$ there are 5350 pairwise non-isomorphic).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-17T17:35:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetrical",
      "realization",
      "positive integer",
      "equivalence",
      "quotient graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 164,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}