{
  "id": "2208.00547",
  "version": "v1",
  "published": "2022-08-01T00:17:31.000Z",
  "updated": "2022-08-01T00:17:31.000Z",
  "title": "All polytopes are coset geometries: characterizing automorphism groups of k-orbit abstract polytopes",
  "authors": [
    "Isabel Hubard",
    "Elías Mochán"
  ],
  "comment": "41 pages, 3 figures",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-know they can be constructed as coset geometries from their automorphism groups. This is also known to be true for 2- and 3- orbit 3-polytopes. In this paper we show that every abstract $n$-polytope can be constructed as a coset geometry. This construction is done by giving a characterization, in terms of generators, relations and intersection conditions, of the automorphism group of a $k$-orbit polytope with given symmetry type graph. Furthermore, we use these results to show that for all $k\\neq 2$, there exist $k$-orbit $n$-polytopes with Boolean groups (elementary abelian 2-groups) as automorphism group, for all $n\\geq 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-01T00:17:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B15",
      "51A10",
      "05E18",
      "06A11"
    ],
    "keywords": [
      "k-orbit abstract polytopes",
      "coset geometry",
      "characterizing automorphism groups",
      "general combinatorial structures",
      "symmetry type graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}