{
  "id": "2304.01079",
  "version": "v1",
  "published": "2023-04-03T15:35:04.000Z",
  "updated": "2023-04-03T15:35:04.000Z",
  "title": "Unitary $L^{p+}$-representations of almost automorphism groups",
  "authors": [
    "Antje Dabeler",
    "Emilie Mai Elkiær",
    "Maria Gerasimova",
    "Tim de Laat"
  ],
  "comment": "4 pages",
  "categories": [
    "math.RT",
    "math.FA",
    "math.GR",
    "math.OA"
  ],
  "abstract": "Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze-Stein property, and let $\\pi$ be a unitary representation of $H$. We show that the representation $\\widetilde{\\pi}$ of $G$ induced from $\\pi$ is an $L^{p+}$-representation if and only if $\\pi$ is an $L^{p+}$-representation. We deduce the following consequence for a large natural class of almost automorphism groups $G$ of trees: For every $p \\in (2,\\infty)$, the group $G$ has a unitary $L^{p+}$-representation that is not an $L^{q+}$-representation for any $q < p$. This in particular applies to the (coloured) Neretin groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-03T15:35:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "automorphism groups",
      "large natural class",
      "open subgroup",
      "locally compact group",
      "neretin groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}