{
  "id": "1703.09037",
  "version": "v1",
  "published": "2017-03-27T12:41:45.000Z",
  "updated": "2017-03-27T12:41:45.000Z",
  "title": "Any Baumslag-Solitar action on surfaces with a pseudo-Anosov element has a finite orbit",
  "authors": [
    "Nancy Guelman",
    "Isabelle Liousse"
  ],
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "We consider $f, h$ homeomorphims generating a faithful $BS(1,n)$-action on a closed surface $S$, that is, $h f h^{-1} = f^n$, for some $ n\\geq 2$. According to \\cite{GL}, after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\\Lambda$ of the action, included in $Fix(f)$. Here, we suppose that $f$ and $h$ are $C^1$ in neighbourhood of $\\Lambda$ and any point $x\\in\\Lambda$ admits an $h$-unstable manifold $W^u(x)$. Using Bonatti's techniques, we prove that either there exists an integer $N$ such that $W^u(x)$ is included in $Fix(f^N)$ or there is a lower bound for the norm of the differential of $h$ only depending on $n$ and the Riemannian metric on $S$. Combining last statement with a result of \\cite{AGX}, we show that any faithful action of $BS(1, n)$ on $S$ with $h$ a pseudo-Anosov homeomorphism has a finite orbit. As a consequence, there is no faithful $C^1$-action of $BS(1, n)$ on the torus with $h$ an Anosov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-27T12:41:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37Bxx",
      "37Exx"
    ],
    "keywords": [
      "finite orbit",
      "pseudo-anosov element",
      "baumslag-solitar action",
      "minimal set",
      "bonattis techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}