I'm working on the following exercise of Jones, Gareth A., and David Singerman. Complex functions: an algebraic and geometric viewpoint. Cambridge university press, 1987.
Exercise: 2D
Let $ S $ be a hyperbolic transformation with fixed-points $ p $ and $ q $. Let $ T $ be a Möbius transformation which maps $ p $ to $ q $. Prove that
- (i) $ STS^{-1}T^{-1} $ is hyperbolic,
- (ii) $ STST^{-1} $ is parabolic.
my attempt for (i)
$STS^{-1}$ is the conjugate transformation of T by S, and this preserves the fixed points of S by Corollar 2.9.4 of the book. Thus, it's hyperbolic.
Then, I stopped as I don't know how to go about from here on.. any help is appreciated!