A claim of at most linear conjugator length growth for split locally compact Euclidean isometry groups is false; the full isometry group of the plane already gives unbounded conjugator lengths.
The geometry of conjugation in Euclidean isometry groups
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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.
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2025 1verdicts
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Conjugator length of locally compact groups of Euclidean isometries
A claim of at most linear conjugator length growth for split locally compact Euclidean isometry groups is false; the full isometry group of the plane already gives unbounded conjugator lengths.