= Axis of a hyperbolic-space isometry
An isometry of <hyperbolic space> with positive attained <translation length> has an invariant complete geodesic joining its two ideal fixed points. This is its axis. It translates along the axis by its translation length; in three dimensions it may additionally rotate around it. In upper half-space, $(z,t)\mapsto(kz,|k|t)$ with $|k|\ne1$ has the vertical axis through zero and length $|\log|k||$. Every connecting path has length at least the change in $\log t$, and a vertical segment attains the bound.
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