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Solvability and Rigidity for Topological Skew Braces

T0 review · 1 major / 2 minor · reviewed 2026-05-20 · grok-4.3

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Pith's one-line read In connected locally compact Hausdorff topological skew braces, solvability of the additive group implies solvability of the multiplicative group.

desk verdict It proves solvability of the additive group implies solvability of the multiplicative group for connected locally compact Hausdorff skew braces, via Lie quotient reduction, plus counterexamples and a rigidity result in the abelian compact case. read the letter →

arxiv 2605.07609 v2 pith:NCCK2JBE submitted 2026-05-08 math.GR math.GN

Marco Damele , Andrea Loi This is my paper · ORCID
classification math.GRmath.GN
keywords topologicalskewbracessolvabilitylocallycompactgroupsconnectedLieaffineactionsrigidityofgroupoperations
checked against Cost.FunctionalEquationembed_strictMono_of_one_lt
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

The paper seeks to determine whether solvability of the additive group law in a topological skew brace necessarily makes the multiplicative group law solvable as well. It establishes that this holds true whenever the skew brace is connected, locally compact, and Hausdorff. A sympathetic reader would care because this provides a topological extension of the finite skew brace solvability problem, showing that the usual counterexamples are blocked by the topological constraints. The authors further establish that each of the three topological conditions is necessary by exhibiting counterexamples in their absence. Additionally, when the skew brace is compact and connected with an abelian additive group, the two operations must be identical.

What carries the argument

Reduction of the additive group to a solvable Lie quotient combined with the affine-action theorem for transitive affine actions having solvable stabilizer identity components.

What would settle it

Finding a connected locally compact Hausdorff topological skew brace in which the additive group is solvable but the multiplicative group is not.

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Extended reading notes

Core claim

If B=(B,·,∘) is a connected locally compact Hausdorff topological skew brace and the additive group (B,·) is solvable, then the multiplicative group (B,∘) is solvable. The proof reduces the additive group to a solvable Lie quotient and invokes an affine-action theorem asserting that a connected Lie group acting transitively and affinely on a connected solvable Lie group with solvable stabilizer identity component must itself be solvable.

Load-bearing premise

The additive group of the skew brace can be reduced to a solvable Lie quotient to which the affine-action theorem applies with a solvable stabilizer.

Editorial extensions

If this is right

  • When the skew brace is also compact and the additive group is abelian, the two group operations coincide.
  • Solvability of the multiplicative group is guaranteed under the stated topological hypotheses.
  • Counterexamples exist if the space is not connected, not locally compact, or not Hausdorff.

Reading between the lines

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

  • Similar transfer results might hold for other group properties such as nilpotency in the same topological setting.
  • The rigidity phenomenon could extend to other algebraic structures on topological groups via analogous Lie quotient reductions.
  • One might test whether the affine-action approach yields parallel conclusions for non-solvable but finite-length groups.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The manuscript proves that if B=(B,·,∘) is a connected locally compact Hausdorff topological skew brace and the additive group (B,·) is solvable, then the multiplicative group (B,∘) is solvable. The proof reduces the additive group to a solvable Lie quotient and applies an affine-action theorem for connected Lie groups acting transitively and affinely on a connected solvable Lie group with solvable stabilizer identity component. Counterexamples demonstrate that connectedness, local compactness, and the Hausdorff property are essential, and a rigidity result shows that the two operations coincide in the compact connected case with abelian additive group.

Significance. If the central claim holds, the result affirmatively resolves the topological analogue of the Byott-Vendramin solvability problem for skew braces in the connected locally compact Hausdorff setting. The Lie-quotient reduction combined with the affine-action theorem provides a structured approach, and the counterexamples usefully delineate the necessity of the topological hypotheses. The rigidity phenomenon in the compact abelian-additive case is a notable strengthening.

major comments (1)
  1. [Proof strategy / reduction to Lie quotient] The reduction step to the solvable Lie quotient B/N (described in the proof strategy) requires explicit verification that N is invariant under the multiplicative operation ∘, so that the transitive affine action descends to the quotient and the cited affine-action theorem applies. The abstract outlines only the additive reduction; without confirmation that the chosen kernel (e.g., radical or maximal compact normal subgroup of (B,·)) is ∘-normal or that the action remains well-defined, the invocation of the theorem on the quotient is not justified.
minor comments (2)
  1. [Notation throughout] Ensure uniform notation for the two operations · and ∘ in all statements of the main theorem and counterexamples.
  2. [Introduction] Add a brief comparison paragraph situating the topological result against the original finite skew-brace solvability theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important point about the justification of the reduction step in the proof. We address the major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Proof strategy / reduction to Lie quotient] The reduction step to the solvable Lie quotient B/N (described in the proof strategy) requires explicit verification that N is invariant under the multiplicative operation ∘, so that the transitive affine action descends to the quotient and the cited affine-action theorem applies. The abstract outlines only the additive reduction; without confirmation that the chosen kernel (e.g., radical or maximal compact normal subgroup of (B,·)) is ∘-normal or that the action remains well-defined, the invocation of the theorem on the quotient is not justified.

    Authors: We agree that an explicit verification of ∘-invariance for the kernel N is necessary for a fully rigorous presentation. In the manuscript, N is taken to be the maximal compact normal subgroup of the additive group (B,·), which is solvable by the structure theory of locally compact groups. The skew brace compatibility condition (a ∘ b) · c = a · (b · (a^{-1} · c)) ensures that left and right multiplications by elements of (B,∘) map normal subgroups of (B,·) to normal subgroups, so N is automatically ∘-invariant. The induced action on the quotient is therefore well-defined and affine. Nevertheless, to address the referee's concern directly, we will insert a short lemma in Section 3 explicitly confirming both the ∘-normality of N and the descent of the transitive affine action to B/N before invoking the cited theorem on connected Lie groups. This will be a minor but clarifying addition. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation reduces to external affine-action theorem on Lie groups

full rationale

The paper's central claim is established by reducing the connected locally compact solvable additive group to a solvable Lie quotient and invoking a cited affine-action theorem stating that a connected Lie group acting transitively and affinely on a connected solvable Lie group with solvable stabilizer identity component is itself solvable. This structure does not involve self-definition of solvability, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the result to unverified prior claims by the same authors. The derivation chain is self-contained once the external theorem and the validity of the quotient reduction are granted; no step equates the output to the input by construction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper relies on standard background results from topological group theory and Lie group theory rather than introducing new free parameters or invented entities.

assumptions (2)
  • standard math Standard facts about solvable Lie groups and their quotients under continuous homomorphisms
    Invoked in the reduction step of the proof strategy described in the abstract.
  • standard math Existence and properties of the affine-action theorem for connected Lie groups
    Cited as the key tool after the Lie quotient reduction.

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Cite this review

Pith. "Pith review of Solvability and Rigidity for Topological Skew Braces." pith.science (2026). https://pith.science/paper/NCCK2JBE

@misc{pith2026260507609,
  author       = {Pith},
  title        = {Pith review of: Solvability and Rigidity for Topological Skew Braces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NCCK2JBE}},
  note         = {Machine review of arXiv:2605.07609}
}
abstract

We study compact and locally compact topological analogues of the Byott--Vendramin solvability problem for finite skew braces, asking whether solvability of the additive group forces solvability of the multiplicative group. Our main theorem proves an affirmative result in the connected locally compact Hausdorff setting: if \(B=(B,\cdot,\circ)\) is a connected locally compact Hausdorff topological skew brace and the additive group \((B,\cdot)\) is solvable, then the multiplicative group \((B,\circ)\) is solvable. The proof proceeds by reducing the additive group to a solvable Lie quotient and then applying an affine-action theorem: a connected Lie group acting transitively and affinely on a connected solvable Lie group, with solvable stabilizer identity component, is itself solvable. We further show that the Hausdorff, local compactness, and connectedness hypotheses are essential by constructing counterexamples when each is omitted. In the compact connected Hausdorff case with abelian additive group, we obtain a stronger rigidity phenomenon: the two group laws coincide.

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Forward citations

Cited by 2 Pith papers

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

  1. A Counterexample to Byott's Conjecture for Finite Skew Braces

    math.GR 2026-07 accept novelty 9.0 of 10

    An explicit finite skew brace with soluble additive group and insoluble multiplicative group, with quotient PSL2(7), disproves Byott's conjecture.

  2. Simple Skew Braces with Cyclic Sylow Subgroups

    math.GR 2026-07 conditional novelty 7.0 of 10

    Finite simple skew braces with cyclic Sylow structure are either trivial, two order-12 exceptions, or have additive group PSL2(p), with a splitting theorem confirming Byott's conjecture for cyclic Sylow 2-subgroups.

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Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 2 Pith papers

  1. [1]

    Auslander,Simply transitive groups of affine motions, Amer

    L. Auslander,Simply transitive groups of affine motions, Amer. J. Math.99(1977), no. 4, 809–826

    work page 1977

  2. [2]

    Baues,Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology43(2004), no

    O. Baues,Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology43(2004), no. 4, 903–924

    work page 2004

  3. [3]

    F. F. Bonsall and J. Duncan,Complete Normed Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 80, Springer-Verlag, Berlin–Heidelberg–New York, 1973

    work page 1973

  4. [4]

    Deré,NIL-affine crystallographic actions of virtually polycyclic groups, Transform

    J. Deré,NIL-affine crystallographic actions of virtually polycyclic groups, Transform. Groups26(2021), no. 4, 1217–1240

    work page 2021

  5. [5]

    Damele and A

    M. Damele and A. Loi,Structural and rigidity properties of Lie skew braces, J. Algebra695(2026), 356–383. doi:10.1016/j.jalgebra.2026.01.048

    work page doi:10.1016/j.jalgebra.2026.01.048 2026

  6. [6]

    Gorshkov and T

    I. Gorshkov and T. Nasybullov,Finite skew braces with solvable additive group, J. Algebra574(2021), 172–183

    work page 2021

  7. [7]

    Skew braces and the Yang-Baxter equa- tion

    L. Guarnieri and L. Vendramin,Skew braces and the Yang–Baxter equation, Math. Comp.86(2017), no. 307, 2519–2534. doi:10.1090/mcom/3161

    work page doi:10.1090/mcom/3161 2017

  8. [8]

    K. H. Hofmann and L. Kramer,Transitive actions of locally compact groups on locally contractible spaces, J. Reine Angew. Math.702(2015), 227–243

    work page 2015

Show all 12 references
  1. [9]

    K. H. Hofmann and S. A. Morris,The Lie Theory of Connected Pro-Lie Groups. A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups, EMS Tracts in Mathematics, vol. 2, European Mathematical Society, Zürich, 2007

    2007

  2. [10]

    K. H. Hofmann and S. A. Morris,The Structure of Compact Groups, 3rd ed., De Gruyter Studies in Mathematics, vol. 25, Walter de Gruyter, Berlin, 2013

    2013

  3. [11]

    Nasybullov,Connections between properties of the additive and the multiplicative groups of a two-sided skew brace, J

    T. Nasybullov,Connections between properties of the additive and the multiplicative groups of a two-sided skew brace, J. Algebra540(2019), 156–167

    2019

  4. [12]

    Smoktunowicz and L

    A. Smoktunowicz and L. Vendramin,On skew braces, J. Comb. Algebra2(2018), no. 1, 47–86. doi:10.4171/JCA/2-1-3 Dipartimento di Matematica, Università di Cagliari (Italy) Email address:m.damele4@studenti.unica.it Dipartimento di Matematica, Università di Cagliari (Italy) Email a...

    2018 doi

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