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  "body": "\n## Summary\n\nA Schottky group is a group of [[Möbius transformations|0SG74WBFA2]] generated by maps that pair off disjoint discs in the [[Riemann sphere|0J78KBPFZB]].\nIt is a Kleinian group that is a [[free group|0KTW26C80J]] and has only loxodromic elements besides the identity.\nThe quotient of its domain of discontinuity is a compact [[Riemann surface|5K0N8QMV6Q]].\nEvery compact Riemann surface arises in this way, which gives a concrete way to build them.\n\n## Explanations\n\n### L1 human\n\nDraw $2g$ disjoint circles on the Riemann sphere and pair them up.\nFor each pair, choose a Möbius map that sends the outside of the first circle onto the inside of the second.\nThe group generated by these $g$ maps moves the region outside all the circles around without overlap.\nGluing each circle to its partner turns that region into a surface with $g$ handles.\nFor $g = 1$ and the map $z \\mapsto 2z$, with circles $\\abs{z} = 1$ and $\\abs{z} = 2$, the glued annulus is a torus.\n",
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