{
  "id": "0705.4054",
  "version": "v1",
  "published": "2007-05-28T15:23:53.000Z",
  "updated": "2007-05-28T15:23:53.000Z",
  "title": "Distortion in Groups of Circle and Surface Diffeomorphisms",
  "authors": [
    "John Franks"
  ],
  "journal": "Panoramas et Synth`eses, Soc. Math.de France {\\bf 21} (2006) 35-52",
  "categories": [
    "math.DS"
  ],
  "abstract": "In these lectures we consider how algebraic properties of discrete subgroups of Lie groups restrict the possible actions of those groups on surfaces. The results show a strong parallel between the possible actions of such a group on the circle $S^1$ and the measure preserving actions on surfaces. Our aim is the study of the (non)-existence of actions of lattices in a large class of non-compact Lie groups on surfaces. A definitive analysis of the analogous question for actions on $S^1$ was carried out by \\'E. Ghys. Our approach is topological and insofar as possible we try to isolate properties of a group which provide the tools necessary for our analysis. The two key properties we consider are almost simplicity and the existence of a distortion element. Both will be defined and described in the lectures. Our techniques are almost all from low dimensional dynamics. But we are interested in how algebraic properties of a group -- commutativity, nilpotence, etc. affect the possible kinds of dynamics which can occur. For most of the results we will consider groups of diffeomorphisms which preserve a Borel probability measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-05-28T15:23:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E30",
      "57S30"
    ],
    "keywords": [
      "surface diffeomorphisms",
      "distortion",
      "algebraic properties",
      "non-compact lie groups",
      "borel probability measure"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.4054F"
    }
  }
}