Let $A$ be a Möbius transformation acting on the Riemann sphere. Show that, if $A$ is not loxodromic, then there is a disc $\Delta$ in the Riemann sphere with $A(\Delta)=\Delta$. Describe all such discs for each Möbius transformation $A$.

Hence, or otherwise, show that the group $G$ of Möbius transformations generated by

$$A: z \mapsto i z \quad \text { and } \quad B: z \mapsto 2 z$$

does not map any disc onto itself.

Describe the set of points of the Riemann sphere at which $G$ acts discontinuously. What is the quotient of this set by the action of $G$ ?