Define three-dimensional hyperbolic space, the translation length of an isometry of hyperbolic 3 -space, and the axis of a hyperbolic isometry. Briefly explain how and why the latter two concepts are related.

Find the translation length of the isometries defined by (i) $z \mapsto k z, k \in \mathbb{C} \backslash\{0\}$ and (ii) $z \mapsto \frac{3 z+2}{7 z+5}$.