MathAudit

Paper metadata

Original submission
May 24, 2024
Latest revision
June 22, 2026 · version 3
Claimed MSC
32G15, 30F40, 20F65, 57K32, 05A30
arXiv categories
math.CV
Comments
19 pages, 10 figures. Updated to the version accepted by Ann. Inst. Fourier

Abstract

We provide simple and provable bounds on the size and shape of the locus of discrete subgroups of PSL(2,C)≅Isom⁡+(H3)\mathsf{PSL}(2,\mathbb{C})\cong\operatorname{Isom}^+(\mathbb{H}^3) which split as a free product of cyclic groups Zp∗Zq\mathbb{Z}_p*\mathbb{Z}_q, 3≤p,q≤∞3\le p,q\le\infty. These sharp bounds meet the fractal boundary of the deformation space in four cusp groups. As an application, we prove a strengthened conjecture of Morier-Genoud, Ovsienko, and Veselov on faithfulness of the specialized Burau representation.

Dependence graphs

Proof lineage

On qq-real and qq-complex numbers

A statement-restricted proof-dependence graph for arXiv:2508.08440v2. It covers the paper's main analytic continuation and convergence theorems, exact $q$-Bessel formulas, monotonicity and almost-everywhere singularity, the unit-circle Jones/Burau bounds, the mod-$m$ classification results, and the hypergeometric construction of $q$-complex numbers. Expansion follows only claims whose truth can affect one of those marked results and stops at published books. Historical citations, comparisons, conjectures, and alternate routes are documented separately as non-edges.

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AI-generated audit

Audit summary

Audited against arXiv v3

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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.

Current report

Detailed mathematical audit

Generated August 27, 2026
01Statements2 reported findingsNo statement issues found

The explicit sharp bounds for the two-generator Kleinian deformation spaces, their cusp-equality cases, and the application to faithful specializations of the reduced Burau representation have the stated parameter scope. The annulus conclusion needed by arXiv:2508.08440 is Corollary 6.5 in current version 3; it was Corollary 6.4 in version 2.

Theorem 6.3 and Corollary 6.5No issue found

The Burau nonfaithfulness locus lies in the claimed annulus

PDF pages 15–17 · arXiv:2405.15970v3

The explicit invariant-domain estimate makes the two elliptic/parabolic generators a discrete free product outside the tangent locus. Residual finiteness and the Hopfian property then make the induced representation faithful. Under the paper's Burau parameter change, this yields the closed annulus R∗≤∣q∣≤R∗−1R_*\le |q|\le R_*^{-1} with R∗=(3−5)/2R_*=(3-\sqrt{5})/2 for all nonfaithful specializations.

Exact accepted-manuscript arXiv version 3 ↗
Theorem 1.1No issue found

The deformation-space bounds retain their elliptic-order hypotheses

PDF pages 1–4 and proofs in Sections 3–6 · arXiv:2405.15970v3

Theorem 1.1 is stated for primitive elliptics of finite orders 3≤p≤q3\le p\le q and keeps the normalization of the trace parameter fixed; parabolic cases are treated separately by limiting arguments rather than silently included in this statement. The equality configurations are the four cusp groups found by the tangent-circle analysis, so the sharpness claim does not assert that every boundary point is a cusp.

02Proofs3 reported findingsNo proof issues found

The trace-coordinate reduction, invariant-disc geometry, tangency optimization, Klein-Maskit combination step, and Hopfian group argument form a complete proof chain. One strict inequality in the prose should include the tangent endpoint, but the preceding lemma contains that endpoint and the downstream focal use is strictly inside the complementary disk.

Proof of Corollary 6.5Suspected typo · an apparently sound repair

A strict boundary sign has a unique harmless repair

PDF pages 16–17 · arXiv:2405.15970v3

The proof prose says the separation argument applies when ∣λ∣>3|\lambda|>\sqrt{3}, although the inequality in Lemma 5.1 includes the tangent case ∣λ∣=3|\lambda|=\sqrt{3} used by Theorem 6.3 to obtain its non-strict bound. Reading the sentence as ∣λ∣≥3|\lambda|\ge\sqrt{3} is therefore forced directly by the preceding lemma and supplies the printed corollary. The focal paper only needs the strict interior zero-free region, so its conclusion is unaffected.

Combination and faithfulness argumentNo issue found in the reviewed proof

The geometric and group-theoretic steps fit together

PDF pages 9–17 · arXiv:2405.15970v3

The disjoint invariant domains meet the hypotheses of the combination theorem and produce the intended free product. The resulting map is surjective onto that free product; because the source is finitely generated and residually finite, the Hopfian step removes a possible kernel. The trace-to-Burau substitution is algebraic and preserves the two reciprocal annulus boundaries.

Versioned locatorSource locator clarified

The focal citation number is stale but version-correct

arXiv:2405.15970v2 Corollary 6.4; arXiv:2405.15970v3 Corollary 6.5

The result cited as Corollary 6.4 in arXiv:2508.08440v2 exists under that number in EGMS version 2. Version 3 inserted an earlier item and renumbered it Corollary 6.5 without changing the annulus statement.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2405.15970v3
Authors listed
Alex Elzenaar, Jinhua Gong, Gaven J. Martin, Jeroen Schillewaert
Audit date
August 27, 2026
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