Paper 3, Section II, D

Let $\mathcal{M}$ be the group of Möbius transformations of $\mathbb{C} \cup\{\infty\}$ and let $\mathrm{SL}_{2}(\mathbb{C})$ be the group of all $2 \times 2$ complex matrices of determinant 1 .

Show that the map $\theta: \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \mathcal{M}$ given by

$\theta\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)(z)=\frac{a z+b}{c z+d}$

is a surjective homomorphism. Find its kernel.

Show that any $T \in \mathcal{M}$ not equal to the identity is conjugate to a Möbius map $S$ where either $S z=\mu z$ with $\mu \neq 0,1$ or $S z=z+1$. [You may use results about matrices in $\mathrm{SL}_{2}(\mathbb{C})$ as long as they are clearly stated.]

Show that any non-identity Möbius map has one or two fixed points. Also show that if $T$ is a Möbius map with just one fixed point $z_{0}$ then $T^{n} z \rightarrow z_{0}$ as $n \rightarrow \infty$ for any $z \in \mathbb{C} \cup\{\infty\}$. [You may assume that Möbius maps are continuous.]

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