Thrice-punctured sphere as a modular quotient ($\mathbb{H}/\Gamma (2)$)

The level-two principal congruence subgroup $\Gamma (2)\subset {\mathrm{PSL}}_2(\mathbb{R})$ 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 $|d\tau |/\mathrm{Im}\tau$ descends to its curvature-minus-one hyperbolic metric.

// Destination: analysis.bigb