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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite centre and no compact factors, of real rank at least $2$.\nLet $\\Gamma$ be an irreducible [[lattice|BTYF5Y5PFY]] in $G$.\nLet $N$ be a [[normal subgroup|0CB0DX5BVV]] of $\\Gamma$.\nThen either $N$ is contained in the [[centre|0DZ93493VN]] of $G$, and so is finite, or $N$ has finite [[index|0DTXBQPA2S]] in $\\Gamma$.\n\n## Remarks\n\nThe proof has two halves, for a non-central $N$.\nOne half shows that $\\Gamma/N$ is amenable, by an analysis of the measurable quotients of the boundary $G/P$, with $P$ a minimal parabolic subgroup.\nThe other half shows that $\\Gamma/N$ has property (T) when $G$ has it; when $G$ does not, a different argument replaces this half.\nA discrete group that is amenable and has property (T) is finite.\n\nExample: $\\Gamma = \\mathrm{SL}_3(\\Z)$.\nThe centre of $\\mathrm{SL}_3(\\R)$ is trivial, so every nontrivial normal subgroup of $\\mathrm{SL}_3(\\Z)$ has finite index.\nThe congruence subgroups, the kernels of reduction modulo $m$, are such normal subgroups.\n\nThe rank condition is necessary.\nThe principal congruence subgroup $\\Gamma(3)$ of $\\mathrm{SL}_2(\\Z)$ is normal, of finite index and free of rank at least $2$.\nIts commutator subgroup $N$ is characteristic in $\\Gamma(3)$, so it is normal in $\\mathrm{SL}_2(\\Z)$.\nBut $N$ is infinite and has infinite index, since $\\Gamma(3)/N$ is free abelian of positive rank.\n\n## Explanations\n\n### L1 human\n\nA normal subgroup lets one form a quotient group.\nFor lattices in higher rank groups, such as $\\mathrm{SL}_3(\\Z)$, every quotient is either finite or almost the whole group.\nSo these groups are almost simple in an infinite sense: they cannot be mapped onto an infinite smaller group.\n\n### L2 human\n\nThe theorem shows that higher rank lattices are just infinite up to the centre.\nIt is a strong contrast to lattices in $\\mathrm{SL}_2(\\R)$, which are often virtually free and have a huge supply of normal subgroups.\nIt works together with the [[Margulis superrigidity theorem|DSKV65NHZZ]] and the [[Margulis arithmeticity theorem|FEJK846N5X]] to describe these lattices completely.\n",
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  "name": "Margulis normal subgroup theorem",
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