{
  "id": "0907.4419",
  "version": "v2",
  "published": "2009-07-25T12:51:41.000Z",
  "updated": "2009-08-14T03:27:47.000Z",
  "title": "Degenerating slopes with respect to Heegaard distance",
  "authors": [
    "Jiming Ma",
    "Ruifeng Qiu"
  ],
  "comment": "V2; Added references and corrected typos",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $M=H_{+}\\cup_{S} H_{-}$ be a genus $g$ Heegaard splitting with Heegaard distance $n\\geq \\kappa+2$: (1) Let $c_{1}$, $c_{2}$ be two slopes in the same component of $\\partial_{-}H_{-}$, such that the natural Heegaard splitting $M^{i}=H_{+}\\cup_{S} (H_{-}\\cup_{c_{i}} 2-handle)$ has distance less than $n$, then the distance of $c_{1}$ and $c_{2}$ in the curve complex of $\\partial_{-}H_{-}$ is at most $3\\mathfrak{M}+2$, where $\\kappa$ and $\\mathfrak{M}$ are constants due to Masur-Minsky. (2) Let $M^{*}$ be the manifold obtained by attaching a collection of handlebodies $\\mathscr{H}$ to $\\partial_{-} H_{-}$ along a map $f$ from $\\partial \\mathscr{H}$ to $\\partial_{-} H_{-}$. If $f$ is a sufficiently large power of a generic pseudo-Anosov map, then the distance of the Heegaard splitting $M^{*}=H_{+}\\cup (H_{-}\\cup_{f} \\mathscr{H})$ is still $n$. The proofs rely essentially on Masur-Minsky's theory of curve complex.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-08-14T03:27:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27"
    ],
    "keywords": [
      "heegaard distance",
      "degenerating slopes",
      "heegaard splitting",
      "curve complex",
      "generic pseudo-anosov map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.4419M"
    }
  }
}