Prove that an orientation-preserving isometry of the ball-model of hyperbolic space $\mathbb{H}^{3}$ which fixes the origin is an element of $S O(3)$. Hence, or otherwise, prove that a finite subgroup of the group of orientation-preserving isometries of hyperbolic space $\mathbb{H}^{3}$ has a common fixed point.

Can an infinite non-cyclic subgroup of the isometry group of $\mathbb{H}^{3}$ have a common fixed point? Can any such group be a Kleinian group? Justify your answers.