Conjugator length of locally compact groups of Euclidean isometries

Petra Schwer; Yuri Santos Rego

REVIEW 2 major objections 4 minor 1 cited by

Conjugator length of locally compact groups of Euclidean isometries

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Pith's one-line read Every split locally compact subgroup of Euclidean isometries has conjugator length growing at most linearly.

desk verdict The main theorem is false: in Isom(R^2), pairs of rotations with word lengths summing to ≤ 3 have conjugators of arbitrarily large translation length, and the proof's uniform pseudoinverse bound fails for O(2). read the letter →

arxiv 2507.01268 v2 pith:QT3M7ATA submitted 2025-07-02 math.GR

Yuri Santos Rego , Petra Schwer This is my paper · ORCID

classification math.GR MSC 20F6522D0520F55

keywords conjugatorlengthlocallycompactgroupsEuclideanisometryCoxetercrystallographicMoore-Penrosepseudoinversegeometricgrouptheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Compactly generated groups come with a word metric: the length of an element is the fewest generators needed to write it. The conjugator length function measures, over all pairs of conjugate elements whose lengths add to at most $n$, the largest shortest length of an element conjugating one to the other. This paper proves that for every split, locally compact subgroup $H$ of the Euclidean isometry group $\operatorname{Isom}(\mathbb{E}^n)$, the conjugator length function grows at most linearly. A split subgroup decomposes as translations semidirect orthogonal transformations, the natural splitting of Euclidean isometries. The theorem adds affine Coxeter groups, split crystallographic groups, and the full isometry group $\operatorname{Isom}(\mathbb{E}^n)$ itself to the list of groups with linear conjugator growth.

What carries the argument

The carrying object is the coconjugation set $C_H(h,h')$, the set of all elements of $H$ that conjugate $h$ to $h'$. Theorem 2.4, restated from a prior result, describes it as a disjoint union, over spherical parts $u$, of affine subspaces of the form $t_{\eta_u + (\operatorname{Fix}(h'_0)\cap L_H)}\,u$, where $\eta_u$ is any solution of the linear system $\lambda' - u\lambda = (I - h'_0)\eta$. To minimize the Euclidean norm of $\eta_u$, the paper takes the Moore–Penrose pseudoinverse $A^+$ of $A = I - h'_0$, whose value $A^+(\lambda' - u\lambda)$ is the minimum-norm real solution; the bounded covolume of the translation lattice $L_H$ then permits passage to a nearby lattice element. The proof of linearity combines a uniform bound on the norms of these pseudoinverses with the quasi-isometry between word length and translation norm from Lemma 3.1.

What would settle it

Calculate the operator norm of $(I - R_\theta)^+$ for $R_\theta \in \mathrm{SO}(2)$ a rotation by angle $\theta$; it equals $1/(2\sin(\theta/2))$ and tends to infinity as $\theta \to 0$. Hence the boundedness claim used to produce $k_1$ is false for the compact spherical subgroup $\mathrm{SO}(2)$, so a proof of linear conjugator length for $L \rtimes \mathrm{SO}(2)$ would need another argument, and any superlinear lower bound for such a group would disprove Theorem 1.1.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: if $H = T_H \rtimes H_0$ is a split, locally compact subgroup of the Euclidean isometry group $\operatorname{Isom}(\mathbb{E}^n)$, then its conjugator length function $\mathrm{CL}_H\colon \mathbb{N} \to \mathbb{Z}_{\ge 0}$ grows at most linearly. The proof first compares word length with the Euclidean norm of the translation part of an element, via the quasi-isometric embedding of the translation subgroup $T_H$ into $\mathbb{R}^n$. It then shows that the shortest possible translation vector of a conjugator is bounded by an affine function of the two translation vectors being conjugated. This is achieved by describing all possible conjugators as solutions of a linear system and using the Moore–Penrose pseudoinverse to locate the minimum-norm solution, combined with compactness of the spherical part $H_0$ and the bounded covolume of the translation lattice $L_H$. Corollary 1.2 records that affine Coxeter groups and split crystallographic groups consequently have linear conjugator length.

Load-bearing premise

The proof needs a single constant $k_1$ bounding the matrix norm of the Moore–Penrose pseudoinverse of $I - h'_0$ for every spherical part $h'_0 \in H_0$; this uniform bound fails when $H_0$ contains rotations by angles arbitrarily close to zero, such as $H_0 = \mathrm{O}(2)$, so the linear estimate in Proposition 3.7 does not cover that case.

Editorial extensions

If this is right

Every split locally compact subgroup of $\operatorname{Isom}(\mathbb{E}^n)$ has conjugator length $O(n)$, including the full isometry group itself.
Affine Coxeter groups and split crystallographic groups have linearly growing conjugator length, as stated in Corollary 1.2.
For nonconstant cases, the minimal conjugator's Euclidean translation norm is bounded by an affine function of the combined translation norms of the two conjugate elements, and this bound is the mechanism behind the word-length bound.
The growth rate of the conjugator length function is independent of the chosen compact generating set for $H$, because any two compact word metrics on a locally compact group are quasi-isometric.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The uniform pseudoinverse bound is the delicate point: for a spherical part such as $H_0 = \mathrm{SO}(2)$, containing rotations arbitrarily close to the identity, the norms of $(I - h'_0)^+$ are unbounded, so the proof's constant $k_1$ cannot be chosen; the theorem for such groups would require a proof that avoids that uniform bound.
If the linear bound survives in the case of a lattice acting by $\mathrm{SO}(2)$, the same argument template may extend to any semidirect product $L \rtimes K$ with $K$ a compact linear group, replacing the uniform pseudoinverse bound by a spectral-gap or discreteness condition on $K$.
A concrete testable consequence of the theorem's mechanism is that in split crystallographic groups the shortest conjugator between two elements of translation length at most $R$ should have translation length $O(R)$, with constants depending only on the coroot lattice and the spherical group; one could compute these constants explicitly in the type $\widetilde{A}_2$ example.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper's main theorem is false, and the proof breaks exactly at the uniform pseudoinverse estimate in Proposition 3.7. In Isom(R^2) with the compact generating set S = {translations of norm ≤ 1} ∪ {rotations by angle ≤ 1}, set ε = 1/M, p = M e_1, h = R_ε, and h' = t_p R_ε t_p^{-1}. Then ℓ_S(h) ≤ 1 and ℓ_S(h') ≤ 2 (the translation part of h' has norm about 1), so the sum is ≤ 3. But any conjugator of h to h' has translation part p, of norm M, so CL(3) is unbounded. No linear (or polynomial) bound exists. This matches the stress-test note exactly.

The problematic step is the claim just before Eq. (3.4) that {|B^+| : B ∈ I − H0} is bounded. It is not. For H0 = O(2) and B_θ = I − R_θ with θ → 0, B_θ is invertible, so B_θ^+ = B_θ^{-1}, and |B_θ^{-1}| = 1/(2 sin(θ/2)) → ∞. The attempted justification using Proposition 2.7 bounds the columns of B1 and B2, but then uses |B2|^{-1} instead of |B2^{-1}|; the inverse blows up as B becomes singular. This is a load-bearing error, not a missing detail.

What is genuinely new: the definition of conjugator length for compactly generated locally compact groups is reasonable, and the idea of using the [MST24] coconjugation description to reduce the problem to linear algebra is sensible. That citation is legitimate, not circular. The paper also contains an internal inconsistency: the arXiv abstract states a trichotomy (zero/linear/unbounded), while the paper's own abstract and Theorem 1.1 claim unconditional linear growth. The trichotomy is closer to the truth, but the body does not prove it.

The salvageable part is the case of finite spherical part H0: then I − H0 is finite, the pseudoinverse norms are bounded, and the linear estimate likely goes through. That would cover affine Coxeter groups and split crystallographic groups and would be a decent short paper.

Recommendation: reject as is; a desk reject with an invitation to resubmit a corrected version (either restricted to finite H0 or with a correct proof of a trichotomy) seems appropriate.

Referee Report

2 major / 4 minor

Summary. The paper defines a conjugator length function for compactly generated locally compact groups and studies split subgroups H = T_H ⋊ H_0 of the full Euclidean isometry group Isom(E^n). Its main result, Theorem 1.1, claims that for every such split locally compact subgroup the conjugator length function CL(n) grows at most linearly. The proof strategy is to reduce CL to a conjugator translation norm CT via quasi-isometry (Lemma 3.1 and Corollaries 3.3–3.4), then to bound the translation part of a shortest conjugator using the coconjugation set description from [MST24] and Moore–Penrose pseudoinverse estimates (Proposition 3.7). The paper also states consequences for affine Coxeter groups and split crystallographic groups, and includes a worked example for the affine Coxeter group of type eA2.

Significance. If Theorem 1.1 were true, it would be a clean and useful extension of known linear conjugator length results to continuous families such as Isom(E^n) itself, and it would recover linear growth for affine Coxeter and split crystallographic groups. The definition of conjugator length for locally compact groups is natural, and the use of the coconjugation description from [MST24] is methodologically appropriate; relying on that prior structural theorem as a black box is not the source of the difficulty. However, the central estimate in Proposition 3.7 is false, and Theorem 1.1 is contradicted by an elementary example in Isom(R^2). The paper therefore does not establish its main claim, and the error is load-bearing rather than a local gap.

major comments (2)

[Proposition 3.7 (paragraph before Eq. (3.3))] The asserted uniform bound on {|B^+| : B ∈ I − H_0} is false. For H_0 = O(2) and B_θ = I − R_θ with 0 < θ small, B_θ is invertible, so B_θ^+ = B_θ^{-1}. Its operator norm is 1/(2 sin(θ/2)), which tends to infinity as θ → 0. Hence no uniform constant k_1 exists. The attempted justification via Proposition 2.7 bounds the columns of B_1 and B_2, but it does not bound B_2^{-1}; as B approaches a singular matrix, B_2^{-1} becomes unbounded. Since k_1 enters the final estimate M_H(x) = k_1 k_3 x + k_0, this invalidates the claimed linear bound in Proposition 3.7.
[Theorem 1.1 (counterexample)] The failure is not merely a proof gap: the conclusion of Theorem 1.1 is false. Take G = Isom(R^2) and S = {t_v : ∥v∥ ≤ 1} ∪ {R_φ : |φ| ≤ 1}. For every M, set ε = 1/M, p = M e_1, h = R_ε, and h' = t_p R_ε t_p^{-1} = t_{(I−R_ε)p} R_ε. Then ℓ_S(h) = ε ≤ 1 and ℓ_S(h') ≤ ⌈∥(I−R_ε)p∥⌉ + 1 ≤ 2 for all M ≥ 1, since ∥(I−R_ε)p∥ = 2M sin(1/(2M)) ≤ 1. Thus ℓ_S(h) + ℓ_S(h') ≤ 3. Any conjugator from h to h' has translation part p: the equation t_v R_φ R_ε R_φ^{-1} t_v^{-1} = h' forces (I − R_ε)v = (I − R_ε)p, and since I − R_ε is invertible for ε not a multiple of 2π, v = p. Because the word metric on G with this compact generating set is quasi-isometric to the displacement of the origin, any element with translation part p has word length at least c∥p∥ = cM for a fixed c > 0. Hence CL(3) ≥ cM for arbitrarily large M, so CL is not linearly bounded and Theorem 1.1 is contradicted.

minor comments (4)

[Abstract and Introduction] The manuscript contains two different abstracts: the article's own abstract says 'grows linearly', while the summary text says 'either has zero growth, grows linearly, or is unbounded'. These are logically incompatible statements, and the wording should be unified.
[Proposition 3.7 and Theorem 1.1] The hypothesis 'nonconstant' in Proposition 3.7 is confusing: Theorem 1.1 claims at most linear growth, and the proof separates constant cases; if CL were unbounded, the proposition would still assert linear growth. The logical structure should be clarified.
[Proof of Lemma 3.1] The quasi-isometry between T_H and H used in Lemma 3.1 is cited to [CdlH16, Proposition 4.C.11]; it would help the reader if the dependence of the constants on the compact set H_0 were made explicit, since the subsequent proof relies heavily on uniformity over H_0.
[Figure 1] The caption refers to yellow and turquoise elements, which may not be distinguishable in black-and-white printing; consider adding shading or labels.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the proof uses an independent structural theorem, and the contentious pseudoinverse bound is a correctness issue, not a circular one.

full rationale

The paper is a pure proof and contains no empirical fits or predictions. The derivation chain is: (1) define CL and CT and show via Lemma 3.1 and Lemma 3.6 that they are quasi-equivalent using standard quasi-isometry arguments; (2) invoke Theorem 2.4 from [MST24] to describe all conjugators as translations along an affine subspace given by Equation (2.3); (3) use Moore-Penrose pseudoinverses to choose the shortest translation vector and bound its norm. No step defines its output in terms of its input: CL is not assumed in bounding CT, and CT is not used to prove Theorem 2.4. The only self-citation is [MST24], a prior structural theorem by one of the present coauthors; it is parameter-free, its assumptions do not include the target linear growth conclusion, and it is used as a black-box structural input rather than being derived from the present claim. The skeptical concern about the uniform boundedness of { |B^+| : B in I-H0 } just before Equation (3.4) is a substantive correctness issue: for H0 = O(2), |(I - R_theta)^+| = 1/(2 sin(theta/2)) is unbounded as theta tends to 0, so the constant k1 may fail to exist. However, that is a mathematical error, not a circularity: the asserted bound does not follow from the inputs by construction, and no fitted parameter is renamed as a prediction. Hence the finding is no significant circularity, with the minor caveat that the central structural input is a same-group citation that is nonetheless independent support under the stated rules.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities. Its proof rests on the structural theorem [MST24] and on compactness/quasi-isometry facts; the main unstated assumptions are the H0-invariance of the lattice and finiteness of CL(n), both of which fail in general for H = Isom(R^2).

assumptions (4)

domain assumption The spherical subgroup H0 is compact and the translation lattice L_H is isomorphic to R^a × Z^b with bounded covolume.
Lemma 2.3, used throughout; follows from local compactness of H and the structure of closed subgroups of R^n.
domain assumption The action of H0 preserves L_H, i.e., u L_H = L_H for all u in H0.
Needed to make Fix(h'_0) rational with respect to L_H so that inequality (3.3) can hold; this invariance is a consequence of H being a subgroup but is never stated or used in the proof.
ad hoc to paper The Moore-Penrose pseudoinverse norm is uniformly bounded over I - H0.
Asserted before Eq. (3.4); false when H0 contains rotations arbitrarily close to identity, e.g., H0 = O(2).
domain assumption The maximum defining CL(n) is finite.
Definition 2.1; not automatic for locally compact groups and false for H = Isom(R^2), where CL(3) is unbounded.

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Cite this review

Pith. "Pith review of Conjugator length of locally compact groups of Euclidean isometries." pith.science (2026). https://pith.science/paper/QT3M7ATA

@misc{pith2026250701268,
  author       = {Pith},
  title        = {Pith review of: Conjugator length of locally compact groups of Euclidean isometries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QT3M7ATA}},
  note         = {Machine review of arXiv:2507.01268}
}

abstract

We consider locally compact subgroups $H$ of the full isometry group $\Isom(\E^n)$ 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 either has zero growth, grows linearly, or is unbounded --- depending on the topology of the spherical part of $H$. Our theorem shows, in particular, that affine Coxeter groups and split crystallographic groups have linear growth for their conjugator length functions.

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Forward citations

Cited by 1 Pith paper

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

Conjugator length in finitely presented groups
math.GR 2026-07 accept novelty 6.0 of 10

The conjugator length function is quadratic for the integral Heisenberg group and Stallings' group, and cyclic-subgroup distortion can be promoted to conjugator length.

Reference graph

Works this paper leans on

12 extracted references · 8 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [5]

[CdlH16] Yves Cornulier and Pierre de la Harpe

arXiv:2506.01248. [CdlH16] Yves Cornulier and Pierre de la Harpe. Metric geometry of locally compact groups, volume 25 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Z¨ urich, 2016.doi:10.4171/166. [CM09] Stephen L. Campbell and Carl D. Meyer. Generalized inverses of linear transformations , vol- ume 56 of Classics in Applied Mathemati...

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[2] [9]

[Sal15] Andrew W

doi:10.1142/S0218196725500109. [Sal15] Andrew W. Sale. The geometry of the conjugacy problem in wreath products and free solvable groups. J. Group Theory , 18(4):587–621,

work page doi:10.1142/s0218196725500109

[3] [1989]

[MST24] Elizabeth Mili´ cevi´ c, Petra Schwer, and Anne Thomas

doi:10.1070/IM1990v035n01ABEH000693. [MST24] Elizabeth Mili´ cevi´ c, Petra Schwer, and Anne Thomas. The geometry of conjugation in Euclidean isometry groups. Preprint,

work page doi:10.1070/im1990v035n01abeh000693

[4] [1995]

[BR25] Martin R

doi:10.1007/978-3-642-61701-0 . [BR25] Martin R. Bridson and Timothy R. Riley. The lengths of conjugators in the model filiform groups. Preprint,

doi:10.1007/978-3-642-61701-0

[5] [2009]

[Lim11] Elon L

doi:10.1137/1.9780898719048. [Lim11] Elon L. Lima. ´Algebra linear. Cole¸ c˜ ao matem´ atica universit´ aria. IMPA, Rio de Janeiro, 8th edition,

work page doi:10.1137/1.9780898719048

[6] [2013]

doi:10.1007/s00039-012-0206-3 . Yuri Santos Rego, University of Lincoln, Charlotte Scott Research Centre for Algebra, Brayford Pool, LN6 7TS Lincoln, UK Email address : ysantosrego@lincoln.ac.uk Petra Schwer, Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 255, 69120 Heidelberg, Germany Email address : schwer@uni-heidelberg.de

doi:10.1007/s00039-012-0206-3

[7] [2014]

[BH99] Martin R

doi:10.1215/ijm/1446819294. [BH99] Martin R. Bridson and Andr´ e Haefliger. Metric spaces of non-positive curvature , volume 319 of Grundlehren der mathematischen Wissenschaften . Springer-Verlag, Berlin,

work page arXiv

[8] [2015]

[Sal16] Andrew W

doi:10.1515/jgth-2015-0009 . [Sal16] Andrew W. Sale. Conjugacy length in group extensions. Comm. Algebra, 44(2):873–897,

work page doi:10.1515/jgth-2015-0009 2015