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  "body": "\n## Summary\n\nA Kleinian group is a discrete subgroup of $\\mathrm{PSL}_2(\\C)$.\nThe group $\\mathrm{PSL}_2(\\C)$ acts on the Riemann sphere by Möbius transformations and on hyperbolic $3$-space by orientation-preserving isometries.\nQuotients of hyperbolic $3$-space by torsion-free Kleinian groups are exactly the complete oriented [[hyperbolic 3-manifolds|GHVW1Z35BS]].\nThe theory links complex analysis, the geometry of $3$-manifolds, and fractal limit sets on the sphere.\n\n## Explanations\n\n### L1 human\n\nMöbius transformations of the sphere, such as $z \\mapsto (az + b)/(cz + d)$, move circles to circles.\nA Kleinian group is a collection of these maps, closed under composition, with no maps arbitrarily close to the identity.\nRepeating the maps of such a group often produces intricate fractal patterns of circles on the sphere.\nThe same maps act as rigid motions of three-dimensional hyperbolic space, and this is how they describe hyperbolic $3$-manifolds.\n",
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