{
  "id": "1810.08847",
  "version": "v1",
  "published": "2018-10-20T19:51:14.000Z",
  "updated": "2018-10-20T19:51:14.000Z",
  "title": "The Mapping Class Group of a Minimal Subshift",
  "authors": [
    "Scott Schmieding",
    "Kitty Yang"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "For a homeomorphism $T \\colon X \\to X$ of a Cantor set $X$, the mapping class group $\\mathcal{M}(T)$ is the group of isotopy classes of orientation-preserving self-homeomorphisms of the suspension $\\Sigma_{T}X$. The group $\\mathcal{M}(T)$ can be interpreted as the symmetry group of the system $(X,T)$ with respect to the flow equivalence relation. We study $\\mathcal{M}(T)$, focusing on the case when $(X,T)$ is a minimal subshift. We show that when $(X,T)$ is a subshift associated to a substitution, the group $\\mathcal{M}(T)$ is an extension of $\\mathbb{Z}$ by a finite group; for a large class of substitutions including Pisot type, this finite group is a quotient of the automorphism group of $(X,T)$. When $(X,T)$ is a minimal subshift of linear complexity satisfying a no-infinitesimals condition, we show that $\\mathcal{M}(T)$ is virtually abelian. We also show that when $(X,T)$ is minimal, $\\mathcal{M}(T)$ embeds into the Picard group of the crossed product algebra $C(X) \\rtimes_{T} \\mathbb{Z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-20T19:51:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mapping class group",
      "minimal subshift",
      "finite group",
      "flow equivalence relation",
      "cantor set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}