Explain briefly how Möbius transformations of the Riemann sphere are extended to give isometries of the unit ball $B^{3} \subset \mathbb{R}^{3}$ for the hyperbolic metric.

Which Möbius transformations have extensions that fix the origin in $B^{3}$ ?

For which Möbius transformations $T$ can we find a hyperbolic line in $B^{3}$ that $T$ maps onto itself? For which of these Möbius transformations is there only one such hyperbolic line?