{
  "id": "2202.01688",
  "version": "v1",
  "published": "2022-02-03T16:55:46.000Z",
  "updated": "2022-02-03T16:55:46.000Z",
  "title": "On upper bounds for the first $\\ell^2$-Betti number",
  "authors": [
    "Carsten Feldkamp",
    "Steffen Kionke"
  ],
  "comment": "10 pages, comments welcome",
  "categories": [
    "math.GR"
  ],
  "abstract": "This article presents a method for proving upper bounds for the first $\\ell^2$-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first $\\ell^2$-Betti number. Our approach extends to generalizations of $\\ell^2$-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-03T16:55:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F05",
      "20F50",
      "20F69"
    ],
    "keywords": [
      "betti number",
      "large prime exponent",
      "q-normal subgroups",
      "burnside groups",
      "proving upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}