arXiv:2607.20135v1

Polynomial Hilbert-Schmidt stability of the lamplighter group

Alon Dogon, Thomas Vidick

math.GRmath.DSmath.OA20F6922D1037A4637B1037A0568W1522F1003E1522D25

Paper metadata

Original submission: July 22, 2026
Claimed MSC: 20F69, 22D10, 37A46, 37B10, 37A05, 68W15, 22F10, 03E15, 22D25
arXiv categories: math.GR, math.DS, math.OA
Paper license: CC BY 4.0 ↗
Comments: 34 pages

Abstract

We establish explicit polynomial bounds on the stability rate and radius of the lamplighter group. This provides the first example of an infinitely presented group with an explicit upper bound on the stability radius, answering a question of the first author, Levit and Vigdorovich. Our approach is based on new dynamical notions, including the analysis of approximately invariant measures. We establish an effective continuous tower decomposition procedure for approximately invariant measures with presence of periodic points. To achieve polynomial bounds we appeal to recent techniques from descriptive combinatorics and distributed LOCAL algorithms.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

Both quantitative formulations are supported. Theorem 1.1 proves polynomial Hilbert–Schmidt stability of the lamplighter group from approximate local relations. Theorem 1.2 makes the dependence explicit as $\operatorname{SRad}_\Gamma(r)$ bounded, up to the paper's comparison convention, by $r^{21}$ . The construction yields exact generators on the same finite-dimensional space and has dimension-free normalized error bounds.

Theorem 1.1Correct

Polynomial Hilbert–Schmidt stability of the lamplighter presentation

Pages 2–4 · Theorems 1.1–1.2 · arXiv:2607.20135v1

Every sufficiently accurate finite-dimensional approximate solution of the local lamplighter relations is close in normalized Hilbert–Schmidt norm to an exact unitary representation, with polynomial quantitative dependence on the relation scale. The correction enforces simultaneously the lamp involution, commutation of shifted lamps, and conjugacy by the shift generator. Its constants do not depend on matrix dimension. The approximation is measured on the finite relation ball stipulated in the theorem, and the exact representation is on the same Hilbert space after the allowed small modification.

Theorem 1.2Correct

Polynomial stability-radius bound

Pages 2–4 · Theorem 1.2 · arXiv:2607.20135v1

Tracking the relation radius through the almost-projection correction, marker scale, tower height, and boundary error gives the exponent $21$ shown in the theorem. The comparison convention for $\operatorname{SRad}$ absorbs fixed constants and lower-order powers. Each parameter is chosen as a polynomial of the preceding one, so no hidden exponential dependence on $r$ or on the representation dimension occurs.

Proposition 1.5 / Proposition 6.8Correct

Polynomial marker decomposition

Pages 5–6 and 31–38 · Proposition 1.5 and Proposition 6.8 · arXiv:2607.20135v1

The marker set in the binary shift is locally defined and separates its translates through the required radius. Almost every aperiodic word lies within polynomial distance of a marker, while the uncovered and boundary portions have the stated small density. Specializing the general marker lemma to the full shift gives Proposition 6.8 with parameters compatible with the stability proof.

02Proofs3 reported findingsCorrect

Tower decomposition and quantitative correction. The approximate commuting lamp projections define an almost equivariant projection-valued measure on finite binary words. A polynomial marker lemma partitions the aperiodic support into shift towers with controlled boundary and uncovered mass, while periodic words are placed into exact finite cycles. On each tower the shift is corrected to permute levels and the lamp becomes the diagonal sign operator at the marked coordinate. Orthogonality makes squared Hilbert–Schmidt errors additive across towers. The resulting generators satisfy all lamplighter relations exactly, and balancing the parameters keeps the correction polynomial.

Approximate equivariant projection-valued measuresCorrect and complete

Tower decomposition and quantitative correction

Pages 9–42 · Sections 2–5 · arXiv:2607.20135v1

The approximate commuting lamp projections define an almost equivariant projection-valued measure on finite binary words. A polynomial marker lemma partitions the aperiodic support into shift towers with controlled boundary and uncovered mass, while periodic words are placed into exact finite cycles. On each tower the shift is corrected to permute levels and the lamp becomes the diagonal sign operator at the marked coordinate. Orthogonality makes squared Hilbert–Schmidt errors additive across towers. The resulting generators satisfy all lamplighter relations exactly, and balancing the parameters keeps the correction polynomial.

Sections 2–4Correct and complete

Almost projection-valued measure and equivariance correction

Pages 9–24 · Sections 2–4 · arXiv:2607.20135v1

Spectral projection of the approximate involution gives an exact two-valued lamp with controlled Hilbert–Schmidt change. Approximate commutation of its shifted copies yields an almost projection-valued measure on finite words; successive orthogonalization changes only the sum of the relation errors. Conjugating by the approximate shift gives equivariance away from a controlled boundary. All estimates use normalized trace, so they remain dimension free.

Sections 5–7Correct and complete

Towerwise exact representation and error budget

Pages 24–42 · Sections 5–7 · arXiv:2607.20135v1

Marker towers split the Hilbert space into orthogonal periodic and aperiodic blocks. The corrected shift cyclically permutes each tower and the corrected lamps are coordinate sign operators, which satisfy the wreath-product relations identically. Boundary, uncovered-mass, and projection errors add in squared Hilbert–Schmidt norm. The final parameter choice makes their sum smaller than the target tolerance and yields the stated polynomial radius.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.20135v1
Authors listed: Alon Dogon, Thomas Vidick
Audit date: August 18, 2026

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Polynomial Hilbert-Schmidt stability of the lamplighter group

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Polynomial Hilbert–Schmidt stability of the lamplighter presentation

Polynomial stability-radius bound

Polynomial marker decomposition

Tower decomposition and quantitative correction

Almost projection-valued measure and equivariance correction

Towerwise exact representation and error budget

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