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  "body": "\n## Summary\n\nA lattice in a [[locally compact group|AHV05ZBCYW]] $G$ is a [[discrete subgroup|5M7D1BSHVZ]] $\\Gamma$ of finite covolume.\nThat is, the quotient $G/\\Gamma$ has finite volume for a $G$-invariant measure.\nIt is cocompact, or uniform, when $G/\\Gamma$ is compact.\nLattices are discrete approximations of $G$; the integer lattice $\\Z^n$ in $\\R^n$ and the modular group $SL_2(\\Z)$ in $SL_2(\\R)$ are the guiding examples.\nThey link group theory with number theory, geometry and ergodic theory.\n\n## Explanations\n\n### L1 human\n\nThe integer points $\\Z^2$ form a grid in the plane, and the grid squares tile the plane by shifts.\nEach tile has area $1$, so the plane \"folded up\" by the grid, which is a torus, has finite area.\nA lattice in a general group is the same idea: a discrete subgroup whose translates of one region of finite volume cover the group.\nThe group $SL_2(\\Z)$ of integer matrices with determinant $1$ sits inside $SL_2(\\R)$ in this way, even though the folded-up space is not compact.\n",
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