Conjugation in Euclidean isometry groups

Surprisingly there is a simple and beautiful description of the geometry of conjugation within any split subgroup H of the full isometry group G of Euclidean space. We prove (yes, also in the talk) with elementary Linear Algebra methods that for any h∈H, the conjugacy class can geometrically be described by a subset of Euclidean space, namely the move-set of its linearisation. Examples of groups H considered include affine Coxeter groups, certain other crystallographic groups, and the group G itself. This is joint work with Elizabeth Mili\'cevi\'c and Anne Thomas.