{
  "id": "2202.08319",
  "version": "v1",
  "published": "2022-02-16T20:20:59.000Z",
  "updated": "2022-02-16T20:20:59.000Z",
  "title": "The dichotomy property of ${\\rm SL}_2(R)$-A short note",
  "authors": [
    "Alexander Alois Trost"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "A recent paper by Polterovich, Shalom and Shem-Tov has shown that non-discrete, conjugation invariant norms on arithmetic Chevalley groups of higher rank give rise to very restricted topologies. Namely, such topologies always have profinite norm-completions. In this note, we sketch an argument showing that this also holds for ${\\rm SL}_2(R)$ for $R$ a ring of algebraic integers with infinitely many units.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-16T20:20:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51Fxx",
      "20-xx"
    ],
    "keywords": [
      "short note",
      "dichotomy property",
      "arithmetic chevalley groups",
      "conjugation invariant norms",
      "higher rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}