{
  "id": "2403.06982",
  "version": "v1",
  "published": "2024-01-10T02:26:11.000Z",
  "updated": "2024-01-10T02:26:11.000Z",
  "title": "Test for amenability for extensions of infinite residually finite groups",
  "authors": [
    "María Isabel Cortez",
    "Jaime Gómez"
  ],
  "comment": "19 pages. arXiv admin note: substantial text overlap with arXiv:2312.12562",
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "Let $G$ be a countable group that admits a minimal equicontinuous action on the Cantor set (for example, a residually finite group). We show that the family of almost automorphic actions of $G$ on the Cantor set, is a test for amenability for $G$. More precisely, we show that if $G$ is non-amenable, then for every minimal equicontinuous system $(Y, G)$, such that $Y$ is a Cantor set, there exists a Toeplitz subshift with no invariant probability measures, whose maximal equicontinuous factor is $(Y,G)$. Consequently, we obtain that $G$ is amenable if and only if every Toeplitz $G$-subshift has at least one invariant probability measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-10T02:26:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "37B10",
      "20E26"
    ],
    "keywords": [
      "infinite residually finite groups",
      "invariant probability measure",
      "cantor set",
      "amenability",
      "extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}