Paper 3, Section II, E

Let $G$ be $S L_{2}(\mathbb{R})$, the groups of real $2 \times 2$ matrices of determinant 1 , acting on $\mathbb{C} \cup\{\infty\}$ by MÃ¶bius transformations.

For each of the points $0, i,-i$, compute its stabilizer and its orbit under the action of $G$. Show that $G$ has exactly 3 orbits in all.

Compute the orbit of $i$ under the subgroup

$H=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d=1\right\} \subset G .$

Deduce that every element $g$ of $G$ may be expressed in the form $g=h k$ where $h \in H$ and for some $\theta \in \mathbb{R}$,

$k=\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$

How many ways are there of writing $g$ in this form?

*Typos? Please submit corrections to this page on GitHub.*