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  "body": "\n## Summary\n\nA lattice in a [[Lie group|7JBRGCVJRB]] $G$ is a [[discrete subgroup|5M7D1BSHVZ]] $\\Gamma$ such that the quotient $G/\\Gamma$ has finite volume for a $G$-invariant measure.\nIt is the case of Lie groups of a [[lattice in a locally compact group|NS2057MFWP]].\nThe lattices in $\\R^n$ are the classical [[lattices|WN9V7WSJKF]] of the geometry of numbers, and $\\mathrm{SL}_n(\\Z)$ is a lattice in $\\mathrm{SL}_n(\\R)$.\nLattices link Lie groups with number theory, through [[arithmetic groups|4XWG6S7R4Z]], and with geometry, as fundamental groups of finite-volume locally symmetric spaces.\n\n## Explanations\n\n### L1 human\n\nA lattice is a discrete set of symmetries of a continuous group that is large enough to fill the group up to a piece of finite size.\nThe integer points $\\Z^2$ in the plane are the model: they are spread out, yet one unit square of the plane already contains a copy of every point modulo $\\Z^2$.\nIn a non-commutative group such as $\\mathrm{SL}_2(\\R)$, the integer matrices $\\mathrm{SL}_2(\\Z)$ play the same part.\nHere the leftover piece is not compact, but it still has finite volume.\n",
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