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 [h]H of h is described geometrically by the move-set of its linearization, while the set of elements conjugating h to a given h'∈ [h]H is described by the fix-set of the linearization of h' . Examples include all affine Coxeter groups, certain crystallographic groups, and the group G itself.