{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "4XQ8TYS2C4"
    }
  ],
  "body": "\n## Summary\n\nLet $G$ be a compactly generated totally disconnected [[locally compact group|AHV05ZBCYW]].\nA Cayley–Abels graph of $G$ is a connected locally finite graph on which $G$ acts vertex-transitively with compact open vertex stabilisers.\nIt generalises the [[Cayley graph|0GYQK9NDJF]] of a finitely generated group, which is the case of a discrete group.\nEvery such $G$ has a Cayley–Abels graph, and any two of them are [[quasi-isometric|6FKYSH729T]].\nSo $G$ has a well-defined large-scale geometry, as in geometric group theory.\n\n## Explanations\n\n### L1 human\n\nFor a finitely generated group, the Cayley graph draws one vertex for each element and joins elements that differ by a generator.\nFor a group like $\\mathrm{PGL}_2(\\Q_p)$ this is not possible with finitely many generators.\nInstead one draws one vertex for each coset of a compact open subgroup $U$, and joins cosets using finitely many extra elements.\nFor $\\mathrm{PGL}_2(\\Q_p)$ with $U = \\mathrm{PGL}_2(\\Z_p)$, one can obtain in this way the regular tree in which each vertex has $p + 1$ neighbours.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22D05",
      "20F65",
      "05C25"
    ],
    "name": "Cayley–Abels graph",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cayley_graph"
      }
    ],
    "tag": "QWG66JE1JE"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/QWG66JE1JE/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/QWG66JE1JE.md",
  "msc": [
    "22D05",
    "20F65",
    "05C25"
  ],
  "name": "Cayley–Abels graph",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QWG/QWG66JE1JE.md",
  "status": null,
  "tag": "QWG66JE1JE",
  "uri": "m2://QWG66JE1JE",
  "url": "https://tamnd.github.io/math2/wiki/cayley-abels-graph/",
  "writing": {
    "tag": "QWG66JE1JE",
    "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
      }
    ]
  }
}