{
  "id": "math/0604612",
  "version": "v2",
  "published": "2006-04-28T02:17:18.000Z",
  "updated": "2007-06-19T20:41:24.000Z",
  "title": "Bounded generation and lattices that cannot act on the line",
  "authors": [
    "Lucy Lifschitz",
    "Dave Witte Morris"
  ],
  "comment": "28 pages, no figures. In addition to minor corrections, one result was simplified (and strengthened) because of a stronger result in the final version of a joint paper with V.Chernousov",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "Let D be an irreducible lattice in a connected, semisimple Lie group G with finite center. Assume that the real rank of G is at least two, that G/D is not compact, and that G has more than one noncompact simple factor. We show that D has no orientation-preserving actions on the real line. (In algebraic terms, this means that D is not right orderable.) Under the additional assumption that no simple factor of G is isogenous to SL(2,R), applying a theorem of E.Ghys yields the conclusion that any orientation-preserving action of D on the circle must factor through a finite, abelian quotient of D. The proof relies on the fact, proved by D.Carter, G.Keller, and E.Paige, that SL(2,A) is boundedly generated by unipotents whenever A is a ring of integers with infinitely many units. The assumption that G has more than one noncompact simple factor can be eliminated if all noncocompact lattices in SL(3,R) and SL(3,C) are virtually boundedly generated by unipotents.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-06-19T20:41:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F60",
      "22E40",
      "57S25"
    ],
    "keywords": [
      "bounded generation",
      "noncompact simple factor",
      "orientation-preserving action",
      "semisimple lie group",
      "finite center"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......4612L"
    }
  }
}