{
  "id": "1709.00769",
  "version": "v1",
  "published": "2017-09-03T21:31:52.000Z",
  "updated": "2017-09-03T21:31:52.000Z",
  "title": "The growth of Betti numbers and approximation theorems",
  "authors": [
    "Steffen Kionke"
  ],
  "comment": "Borel seminar 2017 (14 pages)",
  "categories": [
    "math.AT",
    "math.GT"
  ],
  "abstract": "These short lecture notes provide a brief introduction to the field of homology growth. They are composed out of two lectures, which I have given at the Borel seminar 2017 in Les Diablerets. We give a proof of L\\\"uck's approximation theorem, discuss generalizations and mention some related open problems. Then we discuss the growth of mod-$p$ Betti numbers, where many problems remain open. We take a closer look at the special case of $p$-adic analytic towers and discuss an approximation theorem due to Bergeron-Linnell-L\\\"uck-Sauer and Calegari-Emerton.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-03T21:31:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "approximation theorem",
      "betti numbers",
      "short lecture notes",
      "problems remain open",
      "adic analytic towers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}