Conjugator length of locally compact groups of Euclidean isometries

TL;DR

This paper proves that the conjugator length function for certain subgroups of Euclidean isometries grows linearly, including groups like affine Coxeter and crystallographic groups, revealing a uniform geometric property.

Contribution

It establishes the linear growth of conjugator length functions for a broad class of groups acting on Euclidean space, extending known results to new group families.

Findings

Conjugator length function grows linearly for these groups.

Applicable to affine Coxeter and crystallographic groups.

Includes the full Euclidean isometry group as a special case.

Abstract

We consider locally compact subgroups $H$ of the full isometry group $\mathrm{Isom}(\mathbb{E}^n)$ of Euclidean $n$-space which respect the splitting into an orthogonal and a translation subgroup. We prove that the conjugator length function of such groups grows linearly. Our theorem applies, in particular, to the Lie group $\mathrm{Isom}(\mathbb{E}^n)$ itself but also to affine Coxeter groups and to split crystallographic groups.