{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "E3ECET66DQ"
    },
    {
      "field": "uses",
      "target": "NTK1TVA50A"
    },
    {
      "field": "about",
      "target": "NTK1TVA50A"
    },
    {
      "field": "defines",
      "target": "ZFZR65WYBX"
    }
  ],
  "body": "\n## Summary\n\nA quasi-isometry is a map between [[metric spaces|0VKD1TP7F3]] that distorts distances by at most a fixed factor and a fixed additive error.\nIn addition, its image comes within a bounded distance of every point.\nIt ignores all small-scale structure and keeps only the large-scale shape of a space.\nTwo spaces joined by a quasi-isometry are called quasi-isometric, and this is an equivalence relation.\nIt is the basic notion of sameness in [[geometric group theory|2T3218RXTD]], since all [[word metrics|VCHJVR9NGH]] of a finitely generated group are quasi-isometric to one another.\n\n## Explanations\n\n### L1 human\n\nLook at a space from very far away, so far that bounded errors become invisible.\nTwo spaces are quasi-isometric if they look the same from there.\nThe integers $\\Z$ and the real line $\\R$ are quasi-isometric: the inclusion $\\Z \\to \\R$ keeps distances exactly, and every real number is within $1/2$ of an integer.\nLikewise the grid $\\Z^2$ looks like the plane $\\R^2$.\nA bounded space, such as a disc, is quasi-isometric to a single point.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "20F65",
      "51F30",
      "53C23"
    ],
    "name": "Quasi-isometry",
    "same_as": {
      "wikidata": "Q614527",
      "wikipedia": "Quasi-isometry"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quasi-isometry"
      }
    ],
    "tag": "6FKYSH729T"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/6FKYSH729T/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/6FKYSH729T.md",
  "msc": [
    "20F65",
    "51F30",
    "53C23"
  ],
  "name": "Quasi-isometry",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6FK/6FKYSH729T.md",
  "status": null,
  "tag": "6FKYSH729T",
  "uri": "m2://6FKYSH729T",
  "url": "https://math2.tamnd.com/wiki/quasi-isometry/",
  "writing": {
    "tag": "6FKYSH729T",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}