{
  "id": "2004.11323",
  "version": "v1",
  "published": "2020-04-23T17:17:17.000Z",
  "updated": "2020-04-23T17:17:17.000Z",
  "title": "Maps on the Morse boundary",
  "authors": [
    "Qing Liu"
  ],
  "comment": "27 pages, 9 figures",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "For a proper geodesic metric space $X$, the Morse boundary $\\partial_*X$ focuses on the hyperbolic-like directions in the space $X$. It is a quasi-isometry invariant. That is, a quasi-isometry between two hyperbolic spaces induces a homeomorphism on their boundaries. In this paper, we investigate additional structures on the Morse boundary $\\partial_*X$ which determine $X$ up to a quasi-isometry. We prove that, for $X$ and $Y$ proper, cocompact spaces, a homeomorphism $f$ between their Morse boundaries is induced by a quasi-isometry if and only if $f$ and $f^{-1}$ are bih\\\"older, or quasi-symmetric, or strongly quasi-conformal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-23T17:17:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "57M07"
    ],
    "keywords": [
      "morse boundary",
      "proper geodesic metric space",
      "hyperbolic spaces induces",
      "additional structures",
      "quasi-isometry invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}