= Thrice-punctured sphere as a modular quotient
{title2=$\mathbb H/\Gamma(2)$}
The level-two <principal congruence subgroup> $\Gamma(2)\subset\operatorname{PSL}_2(\mathbb R)$ consists of integer determinant-one matrices congruent to the identity modulo two, modulo the scalar sign. It acts freely and properly discontinuously on the <complex upper half-plane>. An ideal quadrilateral with vertices $-1,0,1,\infty$ has its paired sides identified by $\tau\mapsto\tau+2$ and $\tau\mapsto\tau/(1-2\tau)$. The quotient has genus zero and three punctures, so it is conformally the sphere with three points removed. The invariant metric $|d\tau|/\operatorname{Im}\tau$ descends to its curvature-minus-one <hyperbolic metric>.
// Destination: analysis.bigb
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