{
  "id": "1004.2126",
  "version": "v2",
  "published": "2010-04-13T09:19:50.000Z",
  "updated": "2011-08-22T11:18:40.000Z",
  "title": "Actions of Baumslag-Solitar groups on surfaces",
  "authors": [
    "Nancy Guelman",
    "Isabelle Liousse"
  ],
  "comment": "We clarified and improved some results and we added examples",
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "Let $BS(1,n) =< a, b \\ | \\ aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\\geq 2$. It is known that BS(1,n) is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $. This paper deals with the dynamics of actions of BS(1,n) on closed orientable surfaces. We exhibit a smooth BS(1,n) action without finite orbits on $\\TT ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid. We develop a general dynamical study for faithful topological BS(1,n)-actions on closed surfaces $S$. We prove that such actions $<f,h \\ | \\ h \\circ f \\circ h^{-1} = f^n>$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty. When $S= \\TT^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of BS(1,n) on $\\TT^2$. When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ then $fix(f)$ contains any minimal set.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-08-22T11:18:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37Bxx",
      "37Exx"
    ],
    "keywords": [
      "minimal set",
      "real line",
      "solvable baumslag-solitar group",
      "minimal faithful topological actions",
      "paper deals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.2126G"
    }
  }
}