{
  "id": "2006.07947",
  "version": "v1",
  "published": "2020-06-14T16:28:04.000Z",
  "updated": "2020-06-14T16:28:04.000Z",
  "title": "2-dimensional Coxeter groups are biautomatic",
  "authors": [
    "Zachary Munro",
    "Damian Osajda",
    "Piotr Przytycki"
  ],
  "comment": "16 pages, 12 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $W$ be a $2$-dimensional Coxeter group, that is, a one with $\\frac{1}{m_{st}}+\\frac{1}{m_{sr}}+\\frac{1}{m_{tr}}\\leq 1$ for all triples of distinct $s,t,r\\in S$. We prove that $W$ is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary $W$), and satisfies the fellow traveller property. As a consequence, by the work of Jacek \\'{S}wi\\k{a}tkowski, groups acting properly and cocompactly on buildings of type $W$ are also biautomatic. We also show that the fellow traveller property for the natural language fails for $W=\\widetilde{A}_3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-14T16:28:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "biautomatic",
      "fellow traveller property",
      "natural geodesic language",
      "dimensional coxeter group",
      "natural language fails"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}