{
  "id": "2411.03221",
  "version": "v1",
  "published": "2024-11-05T16:09:56.000Z",
  "updated": "2024-11-05T16:09:56.000Z",
  "title": "Perfect kernel of generalized Baumslag-Solitar groups",
  "authors": [
    "Sasha Bontemps"
  ],
  "comment": "62 pages, 11 figures",
  "categories": [
    "math.GR",
    "math.DS"
  ],
  "abstract": "In this article, we study the space of subgroups of generalized Baumslag-Solitar groups (GBS groups), that is, groups acting cocompactly on an oriented tree without inversion and with infinite cyclic vertex and edge stabilizers. Our results generalize the study of Baumslag-Solitar groups in [CGLMS22]. Given a GBS group G defined by a graph of groups whose existence is given by Bass-Serre theory, we associate to any subgroup of G an integer, which is a generalization of the phenotype defined in [CGLMS22]. This quantity is invariant under conjugation and allows us to decompose the perfect kernel of G into pieces which are invariant under conjugation and on which G acts highly topologically transitively. To achieve this, we interpret graphs of subgroups of G as \"blown up and shrunk\" Schreier graphs of transitive actions of G. We also describe the topology of the pieces which appear in the decomposition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-05T16:09:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E06",
      "20E08"
    ],
    "keywords": [
      "generalized baumslag-solitar groups",
      "perfect kernel",
      "gbs group",
      "infinite cyclic vertex",
      "schreier graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}