The geometry of conjugation in Euclidean isometry groups

Abstract

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 $h\in H$, 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 $h^{\prime }\in [h{]}_H$ is described by the fix-set of the linearization of $h^{\prime }$. Examples include all affine Coxeter groups, certain crystallographic groups, and the group $G$ itself.