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  "body": "\n## Statement\n\nLet $X$ be a compact connected [[Riemann surface|5K0N8QMV6Q]] of [[genus|NJ787GZS1N]] $g \\ge 2$.\nThen the [[automorphism group|0K1SCPBNKJ]] $\\Aut(X)$ of biholomorphic self-maps of $X$ is finite, and its order satisfies\n$$\\abs{\\Aut(X)} \\le 84 (g - 1).$$\n\n## Remarks\n\nFiniteness is a separate step; a standard proof uses the finite set of Weierstrass points of $X$, which every automorphism permutes.\nFor the bound, let $G = \\Aut(X)$ and apply the [[Riemann–Hurwitz formula|42A5ZT6AWN]] to the quotient map $X \\to X / G$, of degree $\\abs{G}$.\nHere $X/G$ has genus $h$, and the branch values have ramification indices $m_1, \\ldots, m_k$.\nIt gives $2g - 2 = \\abs{G} \\left( 2h - 2 + \\sum_{i=1}^{k} (1 - 1/m_i) \\right)$.\nThe bracket is positive, and its smallest positive value is $1/42$, reached for $h = 0$ and $(m_1, m_2, m_3) = (2, 3, 7)$; this gives $\\abs{G} \\le 84(g - 1)$.\nThe hypothesis $g \\ge 2$ is needed: a torus $\\C / \\Lambda$ of genus $1$ has infinitely many automorphisms, the translations, and so does the Riemann sphere.\nExample: the Klein quartic $x^3 y + y^3 z + z^3 x = 0$ has genus $3$ and automorphism group of order $168 = 84 \\cdot 2$, so the bound is attained.\nThe bound is not attained for every genus; for $g = 2$ the largest automorphism group has order $48$.\n\n## Explanations\n\n### L1 human\n\nA surface with at least two holes can have only finitely many symmetries that respect its complex structure.\nHurwitz's theorem gives an explicit ceiling: a surface of genus $g$ has at most $84(g - 1)$ such symmetries.\nThe number $84$ comes from the triangle with angles $\\pi/2$, $\\pi/3$ and $\\pi/7$, the smallest triangle that tiles the hyperbolic plane in this way.\n\n### L2 human\n\nThe theorem is a typical application of the Riemann–Hurwitz formula to a quotient by a group action.\nThe surfaces that reach the bound are called Hurwitz surfaces, and their automorphism groups are the finite quotients of the $(2, 3, 7)$ triangle group, called Hurwitz groups.\nThe same bound holds for smooth projective curves over an algebraically closed field of characteristic $0$, but in positive characteristic it can fail because of wild ramification.\n",
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