{
  "id": "2206.08775",
  "version": "v1",
  "published": "2022-06-17T13:46:30.000Z",
  "updated": "2022-06-17T13:46:30.000Z",
  "title": "Dead ends on wreath products and lamplighter groups",
  "authors": [
    "Eduardo Silva"
  ],
  "comment": "32 pages, 7 figures",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $A\\wr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every standard generating set. This generalizes the case where $B=\\mathbb{Z}$ together with its cyclic generator due to Cleary and Taback. When $B=H*K$ is the free product of two finite groups $H$ and $K$, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of $H$ and $K$. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-17T13:46:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lamplighter group",
      "standard generating set",
      "dead ends",
      "wreath products",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}