Shortest closed geodesic on the three-punctured sphere
= Shortest closed geodesic on the three-punctured sphere
{title2=$\ell_{\min}=2\log(3+2\sqrt2)$}
In curvature $-1$, its length is $2\operatorname{arcosh}3$. The level-two <principal congruence subgroup> has traces congruent to two modulo four, so the smallest hyperbolic absolute <trace> is six, achieved by $\begin{pmatrix}5&2\\2&1\end{pmatrix}$. The <hyperbolic translation length> formula gives the result; the shortest closed <geodesic> is nonsimple.