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  "body": "\n## Summary\n\nA Fuchsian group is a discrete group of orientation-preserving isometries of the hyperbolic plane, that is, a discrete subgroup of $\\mathrm{PSL}_2(\\R)$.\nIt acts properly discontinuously on the [[upper half-plane|X6XGGH5FPW]], and the quotient is a Riemann surface, possibly with cone points and cusps.\nBy the uniformization theorem, every compact Riemann surface of genus at least $2$ arises as such a quotient.\nThe [[modular group|VEKV1EM77B]] and its congruence subgroups are the Fuchsian groups behind modular forms.\n\n## Explanations\n\n### L1 human\n\nThe hyperbolic plane can be drawn as the upper half of the complex plane, where lines are half-circles and the scale shrinks near the bottom edge.\nA Fuchsian group is a set of rigid motions of this plane that is closed under composition and has no motions arbitrarily close to doing nothing.\nIts images of a single tile then cover the plane without crowding, like the famous tilings in Escher's \"Circle Limit\" pictures.\nGluing the edges of one tile gives a surface, and in this way every closed surface with two or more handles gets a hyperbolic shape.\n",
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