The geometry of conjugation in Euclidean isometry groups

We describe the geometry of conjugation within any split subgroup H of the full isometry group G of n-dimensional Euclidean space. We prove that for any h H, the conjugacy class hH of h is described geometrically by the move-set of its linearization, while the set of elements conjugating h to a given h' hH is described by the the fix-set of its linearization. Examples include all affine Coxeter groups, certain crystallographic groups, and the group G itself.