{
  "id": "2308.15376",
  "version": "v1",
  "published": "2023-08-29T15:14:48.000Z",
  "updated": "2023-08-29T15:14:48.000Z",
  "title": "Isoperimetry in Finitely Generated Groups",
  "authors": [
    "Bruno Luiz Santos Correia",
    "Marc Troyanov"
  ],
  "comment": "20 pages. The results in this paper refine those in arXiv:2110.15798",
  "categories": [
    "math.GR"
  ],
  "abstract": "We revisit the isoperimetric inequalities for finitely generated groups introduced and studied by N. Varopoulos, T. Coulhon and L. Saloff-Coste. Namely we show that a lower bound on the isoperimetric quotient of finite subsets in a finitely generated group is given by the $\\U-$transform of its growth function, which is a variant of the Legendre transform. From this lower bound, we obtain some asymptotic estimates for the F{\\o}lner function of the group. The paper also includes a discussion of some basic definitions from Geometric Group Theory and some basic properties of the $\\U$-transform, including some computational techniques and its relation with the Legendre transform.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-29T15:14:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F69",
      "05C63"
    ],
    "keywords": [
      "finitely generated group",
      "isoperimetry",
      "lower bound",
      "legendre transform",
      "geometric group theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}