The geometry of conjugation in Euclidean isometry groups

TL;DR

This paper explores the geometric structure of conjugation in subgroups of Euclidean isometry groups, linking conjugacy classes and conjugators to geometric move-sets and fix-sets of linearizations.

Contribution

It provides a geometric description of conjugacy classes and conjugators in split subgroups of Euclidean isometry groups, including affine Coxeter and crystallographic groups.

Findings

Conjugacy classes are characterized by move-sets of linearizations.

Conjugators are described by fix-sets of linearizations.

Applicable to affine Coxeter, crystallographic groups, and full isometry group.

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