How do you go about showing that a hyperbolic isometry of is determined uniquely by the images of any 3 non-collinear points?
Further more, how can you show from this that every hyperbolic isometry is the product of at most 3 reflections in lines?
Many thanks. x
PS - we are using the half plane model for hyperbolic space.
(1) Just like the Euclidean case --- first show that it must fix the three lines, and extend from there.
(2) Find a line that reflect i to f(i), then it remains to either reflect in a line through f(i) or rotate about f(i), both possible with at most 2 reflections.