{
  "id": "2207.12684",
  "version": "v1",
  "published": "2022-07-26T06:57:30.000Z",
  "updated": "2022-07-26T06:57:30.000Z",
  "title": "Small diameters and generators for arithmetic lattices in $\\mathrm{SL}_2(\\mathbb{R})$ and certain Ramanujan graphs",
  "authors": [
    "Raphael S. Steiner"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT",
    "math.DS",
    "math.GR",
    "math.GT"
  ],
  "abstract": "We show that arithmetic lattices in $\\mathrm{SL}_{2}(\\mathbb{R})$, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $\\mathbb{Q}$, admit a `small' covering set. In particular, we give bounds on the diameter if the quotient space is co-compact. Consequently, we show that these lattices admit small generators. Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-26T06:57:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F06",
      "20H05"
    ],
    "keywords": [
      "ramanujan graphs",
      "arithmetic lattices",
      "small diameters",
      "lattices admit small generators",
      "definite quaternion algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}