{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] and let $\\Gamma$ be a [[discrete subgroup|5M7D1BSHVZ]] of $G$.\nLet $G/\\Gamma$ be the space of left cosets $g\\Gamma$ with the quotient topology, on which $G$ acts by $h \\cdot g\\Gamma = hg\\Gamma$.\nThe subgroup $\\Gamma$ is a *lattice* in $G$ if there is a nonzero [[Radon measure|HZNEQQBCW4]] $\\nu$ on $G/\\Gamma$ such that:\n\n1. $\\nu$ is $G$-invariant: $\\nu(hE) = \\nu(E)$ for every $h \\in G$ and every Borel set $E \\subseteq G/\\Gamma$;\n2. $\\nu$ is finite: $\\nu(G/\\Gamma) \u003c \\infty$.\n\nA lattice $\\Gamma$ is *uniform*, or *cocompact*, if $G/\\Gamma$ is compact.\n\n## Remarks\n\nExample: $\\Z^n$ in $(\\R^n, +)$.\nThe quotient $\\R^n/\\Z^n$ is the torus, and the image of Lebesgue measure on $[0, 1)^n$ is a translation-invariant measure on it of total mass $1$.\nSince the torus is compact, $\\Z^n$ is a uniform lattice.\n\nNon-example: $\\Gamma = \\Z \\times \\set{0}$ in $\\R^2$ is discrete, but fails clause 2.\nThe quotient is the cylinder $(\\R/\\Z) \\times \\R$.\nEvery invariant Radon measure on it is a multiple of the product of the [[Haar measures|VY0HXD3GC4]] of the two factors.\nThe nonzero ones have infinite total mass.\n\nIf $G$ has a lattice, then $G$ is [[unimodular|CXG6JZGMCK]].\nA measurable [[fundamental domain|KNJA2NEZ9F]] for the action of $\\Gamma$ on $G$ by right multiplication has finite Haar measure exactly when $\\Gamma$ is a lattice.\nThe group $SL_2(\\Z)$ is a lattice in $SL_2(\\R)$ that is not uniform.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "NS2057MFWP",
    "kind": "definition",
    "msc": [
      "22D05",
      "22E40",
      "28C10"
    ],
    "name": "Lattice in a locally compact group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup)"
      }
    ],
    "tag": "BW21VJW403",
    "uses": [
      "AHV05ZBCYW",
      "5M7D1BSHVZ",
      "VY0HXD3GC4",
      "HZNEQQBCW4",
      "CXG6JZGMCK",
      "KNJA2NEZ9F"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/BW21VJW403/",
  "links": [
    {
      "field": "defines",
      "target": "NS2057MFWP"
    },
    {
      "field": "uses",
      "target": "AHV05ZBCYW"
    },
    {
      "field": "uses",
      "target": "5M7D1BSHVZ"
    },
    {
      "field": "uses",
      "target": "VY0HXD3GC4"
    },
    {
      "field": "uses",
      "target": "HZNEQQBCW4"
    },
    {
      "field": "uses",
      "target": "CXG6JZGMCK"
    },
    {
      "field": "uses",
      "target": "KNJA2NEZ9F"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/BW21VJW403.md",
  "msc": [
    "22D05",
    "22E40",
    "28C10"
  ],
  "name": "Lattice in a locally compact group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BW2/BW21VJW403.md",
  "status": null,
  "tag": "BW21VJW403",
  "uri": "m2://BW21VJW403",
  "url": "https://tamnd.github.io/math2/wiki/lattice-in-a-locally-compact-group-definition/",
  "writing": {
    "tag": "BW21VJW403",
    "kind": "definition",
    "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
      }
    ]
  }
}