{
  "id": "1312.7418",
  "version": "v1",
  "published": "2013-12-28T10:30:14.000Z",
  "updated": "2013-12-28T10:30:14.000Z",
  "title": "Centralizers of $C^1$-contractions of the half line",
  "authors": [
    "Christian Bonatti",
    "Églantine Farinelli"
  ],
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "A subgroup $G\\subset Diff^1_+([0,1])$ is $C^1$-close to the identity if there is a sequence $h_n\\in Diff^1_+([0,1])$ such that the conjugates $h_n g h_n^{-1}$ tend to the identity for the $C^1$-topology, for every $g\\in G$. This is equivalent to the fact that $G$ can be embedded in the $C^1$-centralizer of a $C^1$-contraction of $[0,+\\infty)$ (see [Fa] and Theorem 1.1). We first describe the topological dynamics of groups $C^1$-close to the identity. Then, we show that the class of groups $C^1$-close to the identity is invariant under some natural dynamical and algebraic extensions. As a consequence, we can describe a large class of groups $G\\subset Diff^1_+([0,1])$ whose topological dynamics implies that they are $C^1$-close to the identity. This allows us to show that the free group ${\\mathbb F}_2$ admits faithfull actions which are $C^1$-close to the identity. In particular, the $C^1$-centralizer of a $C^1$-contraction may contain free groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-28T10:30:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22F05",
      "37C85"
    ],
    "keywords": [
      "half line",
      "contraction",
      "centralizer",
      "admits faithfull actions",
      "contain free groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.7418B"
    }
  }
}