{
  "id": "2411.04250",
  "version": "v1",
  "published": "2024-11-06T20:42:40.000Z",
  "updated": "2024-11-06T20:42:40.000Z",
  "title": "A Tits alternative for $\\mathbb{R}$-buildings of type $\\tilde{A}_2$",
  "authors": [
    "Corentin Le Bars",
    "Jean Lécureux",
    "Jeroen Schillewaert"
  ],
  "comment": "25 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a group with a non-elementary action on a (not necessarily discrete) $\\tilde{A}_2$-buildings. We prove that, given a random walk on $G$, isometries in $G$ are strongly regular hyperbolic with high probability. As a consequence, we prove a Tits alternative for $G$, as well as a local-to-global fixed point result. We also prove that isometries of (not necessarily complete) $\\mathbb{R}$-buildings are semi-simple.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-06T20:42:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20E42",
      "60G50"
    ],
    "keywords": [
      "tits alternative",
      "local-to-global fixed point result",
      "isometries",
      "non-elementary action",
      "high probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}