= Real determinant-one Möbius orbits
{title2=$\mathbb R\cup\{\infty\},\ \operatorname{Im}z>0,\ \operatorname{Im}z<0$}
The <special linear group> $SL_2(\mathbb R)$ acting by <Möbius transformations> has exactly three <group orbits> on the <Riemann sphere>: the extended real line and the two open half-planes. The identity $\operatorname{Im}((az+b)/(cz+d))=\operatorname{Im}z/|cz+d|^2$ preserves these sets. Upper triangular determinant-one matrices send $i$ to any prescribed point of the <complex upper half-plane>, and similarly send $-i$ through the lower half-plane. The <stabilizer subgroups> of $i$ and $-i$ are both $SO(2)$.
Back to article page
