= Solution
A model of three-dimensional <hyperbolic space> is $\mathbb H^3=\{(z,t):z\in\mathbb C,t>0\}$ with metric $ds^2=(|dz|^2+dt^2)/t^2$, the complete <simply connected> metric of curvature $-1$. The <translation length> of an <isometry> $g$ is $\ell(g)=\inf_Pd(P,gP)$. A hyperbolic <isometry> has positive <translation length> attained on its invariant complete <geodesic>, the <axis of a hyperbolic-space isometry>. It joins the two fixed ideal points and is translated by distance $\ell(g)$; a three-dimensional axial <isometry> can also rotate around it.
For the boundary map $z\mapsto kz$, the <hyperbolic extension of a Möbius transformation> is $(z,t)\mapsto(kz,|k|t)$. Every path from $P$ to $gP$ has length at least the absolute change of $\log t$, namely $|\log|k||$, because the metric length dominates $|dt|/t$. On the vertical line $z=0$ a vertical segment attains that bound. Hence
$$
\boxed{\ell(g)=|\log|k||.}
$$
If $|k|\ne1$ the vertical line is the axis. If $|k|=1$ the map is a rotation with zero <translation length>, including the identity when $k=1$; it is not a positive-length hyperbolic <isometry>.
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