{
  "id": "1208.1023",
  "version": "v1",
  "published": "2012-08-05T15:33:01.000Z",
  "updated": "2012-08-05T15:33:01.000Z",
  "title": "Subgroup of interval exchanges generated by torsion elements and rotations",
  "authors": [
    "Michael Boshernitzan"
  ],
  "comment": "8 pages",
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "Denote by $G$ the group of interval exchange transformations (IETs) on the unit interval. Let $G_{per}\\subset G$ be the subgroup generated by torsion elements in $G$ (periodic IETs), and let $G_{rot}\\subset G$ be the subset of 2-IETs (rotations). The elements of the subgroup $G_1=< G_{per},G_{rot}>\\subset G$ (generated by the sets $G_{per}$ and $G_{rot}$) are characterized constructively in terms of their Sah-Arnoux-Fathi (SAF) invariant. The characterization implies that a non-rotation type 3-IET lies in $G_1$ if and only if the lengths of its exchanged intervals are linearly dependent over $\\Q$. In particular, $G_1\\subsetneq G$. The main tools used in the paper are the SAF invariant and a recent result by Y. Vorobets that $G_{per}$ coincides with the commutator subgroup of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-05T15:33:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "20Exx",
      "54H15"
    ],
    "keywords": [
      "interval exchanges",
      "torsion elements",
      "interval exchange transformations",
      "saf invariant",
      "main tools"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.1023B"
    }
  }
}