Conjugator length of locally compact groups of Euclidean isometries

We consider locally compact subgroups $H$ of the full isometry group $\Isom(\E^n)$ of Euclidean $n$-space which respect the splitting into an orthogonal and a translation subgroup.

We prove that the conjugator length function of such groups either has zero growth, grows linearly, or is unbounded --- depending on the topology of the spherical part of $H$.

Our theorem shows, in particular, that affine Coxeter groups and split crystallographic groups have linear growth for their conjugator length functions.