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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] and let $\\Gamma$ be a [[discrete subgroup|5M7D1BSHVZ]] of $G$.\nThe group $G$ acts on the coset space $G/\\Gamma$ by left multiplication.\nThe subgroup $\\Gamma$ is a *lattice* in $G$ if there is a nonzero finite Borel measure on $G/\\Gamma$ that is invariant under this action.\nA lattice $\\Gamma$ is *uniform*, or *cocompact*, if $G/\\Gamma$ is compact, and *non-uniform* otherwise.\n\n## Remarks\n\nWhen such a measure exists, it is unique up to a positive factor and is induced by a [[Haar measure|VY0HXD3GC4]] $\\mu$ of $G$.\nThe total mass of $G/\\Gamma$ for the measure induced by $\\mu$ is the *covolume* of $\\Gamma$.\nA Lie group that contains a lattice is unimodular.\nExample: $\\Z^n$ is a uniform lattice in $\\R^n$, because $\\R^n/\\Z^n$ is a compact torus whose Lebesgue measure is finite and invariant under translations.\nExample: $\\mathrm{SL}_2(\\Z)$ is a non-uniform lattice in $\\mathrm{SL}_2(\\R)$; the modular surface has finite hyperbolic area $\\pi/3$ but is not compact.\nNon-example: $\\Z \\times \\set{0}$ is discrete in $\\R^2$, but $\\R^2/(\\Z \\times \\set{0})$ is an infinite cylinder, and any invariant measure is a multiple of its infinite area measure.\nNon-example: the subgroup $\\Z + \\sqrt{2}\\,\\Z$ of $\\R$ is dense, not discrete, so it is not a lattice.\n",
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