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The geometry of conjugation in Euclidean isometry groups

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arxiv 2407.08078 v3 pith:ORK4SN35 submitted 2024-07-10 math.GR math.GT

classification math.GRmath.GT
keywords groupsconjugationdescribedeuclideangeometrygroupisometrylinearization
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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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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Conjugator length of locally compact groups of Euclidean isometries

    math.GR 2025-07 reject novelty 6.0 of 10

    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.

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