{
  "id": "2401.06886",
  "version": "v1",
  "published": "2024-01-12T20:38:34.000Z",
  "updated": "2024-01-12T20:38:34.000Z",
  "title": "On the growth of actions of free products",
  "authors": [
    "Le Boudec Adrien",
    "Matte Bon Nicolás",
    "Salo Ville"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "If $G$ is a finitely generated group and $X$ a $G$-set, the growth of the action of $G$ on $X$ is the function that measures the largest cardinality of a ball of radius $n$ in the Schreier graph $\\Gamma(G,X)$. In this note we consider the following stability problem: if $G,H$ are finitely generated groups admitting a faithful action of growth bounded above by a function $f$, does the free product $G \\ast H$ also admit a faithful action of growth bounded above by $f$? We show that the answer is positive under additional assumptions, and negative in general. In the negative direction, our counter-examples are obtained with $G$ either the commutator subgroup of the topological full group of a minimal and expansive homeomorphism of the Cantor space; or $G$ a Houghton group. In both cases, the group $G$ admits a faithful action of linear growth, and we show that $G\\ast H$ admits no faithful action of subquadratic growth provided $H$ is non-trivial. In the positive direction, we describe a class of groups that admit actions of linear growth and is closed under free products and exhibit examples within this class, among which the Grigorchuk group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-12T20:38:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free product",
      "faithful action",
      "finitely generated group",
      "linear growth",
      "admit actions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}