The geometry of conjugation in Euclidean isometry groups

Document Type

Journal Article

Role

Author

Published In

L’Enseignement Mathématique

Volume

72

Issue

3/4

First Page

447

Last Page

466

Publication Date

9-30-2025

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∈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.

Comments

Preprint version available at https://doi.org/10.48550/arXiv.2407.08078

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