Inducing spectral gaps for the cohomological Laplacians of $\operatorname{Sp}_{2n}(\mathbb{Z})$

Jakub Szyma\'nski; Piotr Mizerka

REVIEW 4 major objections 5 minor 22 references

Inducing spectral gaps for the cohomological Laplacians of $\operatorname{Sp}_{2n}(\mathbb{Z})$

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves that a sum-of-squares certificate for one block of the cohomological Laplacian of the integral symplectic group at rank m induces a scaled certificate at every higher rank, turning a finite computation into uniform…

desk verdict A useful Adj-block induction for Sp_{2n}(Z), with an over-advertised main theorem that needs fixing before the paper is accepted. read the letter →

arxiv 2504.16625 v1 pith:AF4XPURM submitted 2025-04-23 math.GR math.FAmath.OA

Piotr Mizerka , Jakub Szymański This is my paper · ORCID

classification math.GRmath.FAmath.OA MSC 20F6520G3520J0622D10

keywords spectralgapcohomologicalLaplaciansumofsquaressymplecticgroupSp(2nZ)rankinductionproperty(T)KazhdanconstantSteinbergpresentation

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 establishes a rank-induction mechanism for the first cohomological Laplacian of the integral symplectic groups $\operatorname{Sp}_{2n}(\mathbb{Z})$. It shows that if the adjacency-type summand $Adj_m$ is a sum of squares up to a margin $\lambda$ at rank $m$, then the corresponding summand at every larger rank $n$ is automatically a sum of squares up to the scaled margin $((n-2)/(m-2))\lambda$. The same mechanism works in a weaker form when only $Adj^-_m + \Delta^+_1$ is certified, adding a large multiple of $\Delta^+_1$ at higher ranks. Applied to computer-certified rank-3 base checks, this yields explicit lower bounds for the Laplacian spectral gap of every quotient $H_n = \operatorname{Sp}_{2n}(\mathbb{Z})/\langle [Z_i,Z_i']\rangle$, together with direct certificates for $\operatorname{Sp}_4(\mathbb{Z})$ and $\operatorname{Sp}_6(\mathbb{Z})$. A reader should care because the method converts one finite computation into infinitely many certified inequalities, without solving larger semidefinite problems at each rank.

What carries the argument

The load-bearing object is the decomposition of $\Delta_1$ into four summands indexed by the size of the set of indices appearing in a generator or relator: $Mono$ (one index), $Sq$ (two), $Adj$ (three), and $Op$ (four). For the $Adj$ summand at rank $m$, the group of index permutations $\operatorname{Sym}_m$ acts on the triangles that index its entries, so the orbit-stabilizer theorem expresses the rank-$n$ block as an explicit normalized sum of the embedded rank-$m$ block over all $n!$ permutations. Lemma 4.2 computes the symmetrized identity matrix as a combination of the rank-$n$ identity split into its $Mono$ and $Sq$ diagonal blocks, with explicit combinatorial coefficients. Substituting these two identities into $Adj_m - \lambda I_m \ge 0$ produces the rank-$n$ certificate with an extra non-negative term, which is why the scaled margin has the factor $(n-2)/(m-2)$.

What would settle it

Recompute the two rank-3 quantities in exact rational arithmetic, or with a second independently written certifier, and check whether the sum-of-squares decompositions asserted in Lemma 5.2 are reproduced; a single negative eigenvalue in the certified matrix would refute the base case and, with it, the induced bounds for every quotient $H_n$.

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Extended reading notes

Core claim

The central claim is Theorem 4.3. Write $RG$ for the group ring and interpret $M \ge 0$ to mean that $M$ is a sum of matrices of the form $N^*N$. The paper proves that whenever $Adj_m - \lambda I_m \ge 0$ in $M_{2m^2 \times 2m^2}(RG_m)$ for some $m \ge 2$, then $Adj_n - ((n-2)/(m-2))\lambda I_n \ge 0$ in $M_{2n^2 \times 2n^2}(RG_n)$ for every $n \ge m$. The proof averages the rank-$m$ identity over all permutations of the $n$ indices, using an orbit-stabilizer count that turns the symmetrized $Adj_m$ into $Adj_n$ and the symmetrized identity matrix into the identity plus an explicit positive correction term. A second statement shows that if only the partial certificate $Adj^-_m + \Delta^+_1 - \lambda I_m \ge 0$ is available, the same averaging gives $Adj^-_n + k\Delta^+_1 - ((n-2)/(m-2))\lambda I_n \ge 0$ for sufficiently large $k$. The paper couples this theorem with structural identities ($Mono^-_n$ and $Op^+_n + Op^-_n$ are always sums of squares) and with two certified rank-3 identities, deriving the uniform bounds for the quotients $H_n$ and the explicit constants $0.82$ and $0.99$ for $\operatorname{Sp}_4(\mathbb{Z})$ and $\operatorname{Sp}_6(\mathbb{Z})$.

Load-bearing premise

The load-bearing premise is that the two rank-3 computer certificates in Lemma 5.2—that $Adj^-+\Delta^+_1-0.24I_3$ and $Sq^-+\Delta^+_1-0.99I_3$ are sums of squares—are correct and that the interval-arithmetic certification turning numerical output into exact inequalities is sound; if either certificate fails, the induced bounds for all higher ranks collapse.

Editorial extensions

If this is right

A rank-$m$ certificate for $Adj_m$ yields, for every $n \ge m$, a certified lower bound $((n-2)/(m-2))\lambda$ for $Adj_n$ without solving a new semidefinite program at rank $n$.
The rank-3 certificates of Lemma 5.2 imply, for every $n \ge 2$, that $\Delta^-_1 + k\Delta^+_1 - ((n-2)0.24 + 0.99)I_n$ is a sum of squares in $M_{2n^2 \times 2n^2}(R H_n)$, with $k$ sufficiently large.
Direct computations give $\Delta_1 - 0.82 I_2 \ge 0$ for $\operatorname{Sp}_4(\mathbb{Z})$ and $\Delta_1 - 0.99 I_3 \ge 0$ for $\operatorname{Sp}_6(\mathbb{Z})$.
Via Remark 1.1, these direct certificates also give lower bounds $0.82$ and $0.99$ for the spectral gap in the Ozawa expression $\Delta^2 - \lambda\Delta$ for $\operatorname{Sp}_4(\mathbb{Z})$ and $\operatorname{Sp}_6(\mathbb{Z})$.
The structural identities $Mono^-_n \ge 0$ and $Op^+_n + Op^-_n \ge 0$ hold for every $n$, so the computational burden is confined to the $Sq$ and $Adj$ summands.

Reading between the lines

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

The symmetrization argument uses only the invariance of generators and relators under index permutations, not the specific arithmetic of $\operatorname{Sp}(2n,\mathbb{Z})$; the same orbit-stabilizer computation should transfer to any group family with a simplex-indexed presentation and an $Adj$-type summand. This is my inference, not a claim of the paper.
The factor $(n-2)/(m-2)$ rewards larger base ranks: a certificate at $m=4$ or $5$ would improve the induced lower bound for all higher $n$ at no additional structural work, making the cost of the base semidefinite program the real bottleneck. This is an editorial consequence of the theorem.
Because the certified bound grows linearly in $n$ while the generating set has size $2n^2$, the induced Kazhdan-constant lower bound decays roughly like $n^{-1/2}$; a natural testable extension is whether the true optimal gap decays at the same rate for the quotients $H_n$. This is my inference.

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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

The useful thing here is the rank induction for the Adj block of the cohomological Laplacian of Sp_{2n}(Z). That is new in this setting, and after you repair a couple of normalization errors the argument goes through. The symmetrization lemmas are a clean extension of Mizerka's work for SL_n and SAut(F_n) and the Kaluba–Kielak–Nowak approach; the decomposition into Mono/Sq/Adj/Op is the right way to see why the Adj part symmetrizes at the right rate.

The advertised central claim is not what is proved. The abstract and Theorem 0.2 say an Adj_m sum-of-squares bound gives a Delta_1 spectral gap for all n >= m. Section 4 only proves an Adj_n bound; nothing there controls Sq, Mono, or Op. The stress-test note is correct on this. The application in Section 5 needs a separate computer certificate for Sq^- + Delta^+_1 (Lemma 5.2) and nonnegativity of Mono^- and Op^+ + Op^- (Lemma 5.1). So Theorem 0.2 should be rewritten as an Adj-block induction, and the abstract should say that the full Delta_1 gap is assembled from additional inputs. This is a fixable overstatement, not a fatal flaw.

There are also small technical errors in Theorem 4.3 as written. The coefficient in the symmetrized identity is wrong: it should be (m-3)!/((n-3)!*m!), not the printed (n-2)! and m! factor, so the hypothesis range m >= 2 should be m >= 3. The displayed identity with I^Mono is false for lambda != 1; the correct computation adds a nonnegative lambda (n-2)(n-m)/((m-2)(m-1)) I^Mono term, so the desired inequality still follows. The n = 2 case of Theorem 5.4 is not covered by the m = 3 induction; it needs a separate argument, possibly from the G_2 computation, though that yields a different bound.

On the computational side, Lemma 5.2 is not independently checked here, but the repository and interval-arithmetic setup are provided, and the certification method in Appendix B is a genuine small improvement. I did not rerun the certificates, so I treat them as probable but not fully verified by me.

Who is this for? People in the computational property (T) program, especially those working on Sp-groups or rank induction for cohomological Laplacians. It is a solid extension of an established line. I would send it to a serious referee and make the authors fix the abstract, Theorem 0.2, and the normalization before acceptance. The main application appears salvageable with those changes.

Referee Report

4 major / 5 minor

Summary. The paper develops an induction technique for the first cohomological Laplacian of Sp_{2n}(Z), seeking to infer a spectral gap at rank n from a sum-of-squares certificate at some smaller rank m. The main advertised result (Theorem 0.2) claims that a sum-of-squares decomposition for the summand Adj at rank m implies a spectral gap for the full Laplacian Δ1 at every rank n ≥ m. The actual proof in Section 4 establishes, at most, an induction statement for the Adj summand alone. The paper also contains a computer-assisted application: using base certificates for H3, it claims explicit lower bounds for the spectral gap of certain quotients H_n for all n ≥ 2. The computational material includes rigorous certification via interval arithmetic and a Wedderburn-decomposition acceleration.

Significance. If the induction were established in the advertised form, it would be a useful rank-reduction tool for cohomological Laplacians of symplectic groups, complementing the existing methods of Kaluba–Kielak and the first author's earlier work on SL_n(Z). The paper's strengths are its reproducible computational framework, the use of rigorous interval-arithmetic certificates, and the explicit constants. However, the significance is currently undermined because the main theorem as stated does not follow from the proof: the proof only concerns the Adj block, and the transition from Adj to Δ1 requires additional, separately certified summands. The corrected theorem would be narrower than the abstract claims, and the proof of the application still needs substantial justification.

major comments (4)

[Section 4.4, proof of Theorem 4.3] The displayed identity after Lemma 4.2 is false. With C = (m−3)!/((n−3)!·m!), Lemma 4.2 gives C∑σ σ(I_m) = C[m(n−1)! I_n^Mono + m(m−1)(n−2)! I_n^Sq] = [(n−1)(n−2)/((m−1)(m−2))] I_n^Mono + [(n−2)/(m−2)] I_n^Sq. Therefore C∑σ σ(Adj_m − λI_m) differs from Adj_n − ((n−2)/(m−2))λI_n, and adding ((n−m)/(m−1))I_n^Mono does not repair the equality. For m = 3, n = 4, λ = 1 the coefficient of I_n^Mono in C∑σ σ(I_m) is 3, while the target identity requires coefficient 2. The induction step therefore is not proved as written. A corrected inequality appears to be salvageable for m ≥ 3 because the excess on I_n^Mono is nonnegative, but this needs to be derived explicitly.
[Theorem 0.2 and Section 4.4] Theorem 0.2 and the abstract assert that a sum-of-squares certificate for Adj_m − λI_m implies that Δ1 − ((n−2)/(m−2))λI_n is a sum of squares for all n ≥ m. The proof of Theorem 4.3 establishes, at best, an inequality for the Adj_n summand; no argument in Section 4 controls Sq^−_n, Mono^−_n, or Op^±_n. The later application confirms this gap: Theorem 5.4 needs the additional computer-assisted certificate Sq^− + Δ1^+ − 0.99I_3 and Lemma 5.1 to assemble the full Δ1. If Theorem 0.2 were true as stated, those inputs would be redundant. The authors should either prove the full Δ1 statement or revise the abstract and introduction to announce the Adj-block induction that is actually shown.
[Theorem 5.4, proof after Lemma 5.2] The assertion 'Similarly, by Lemmas 4.1 and 4.2 and Lemma 5.2, we get Sq^−_n + Δ1^+ − 0.99I_n ≥ 0' is not justified. Lemma 4.1 applies to a single summand such as Sq^− (where the simplex size is k = 2), while the expression Sq^−_3 + Δ1^+ − 0.99I_3 mixes terms of support sizes 1 through 4. Symmetrizing this whole expression is not covered by the stated lemmas, and the multiplicity of the Δ1^+ contribution under symmetrization is not computed. Consequently the proof of Theorem 5.4 does not follow from the displayed ingredients.
[Theorem 5.4, range n ≥ 2] The induction base in Lemma 5.2 is at m = 3, and Theorem 4.3 supplies conclusions only for n ≥ m. The theorem therefore yields statements for n ≥ 3, not for n ≥ 2 as claimed. The case n = 2 would require a separate base at m = 2 for the relevant quotient, but the coefficient ((n−2)/(m−2)) is undefined at m = 2; this case is not treated in the proof. The authors should either prove the n = 2 case separately or restrict the range in Theorem 5.4.

minor comments (5)

[Section 4.2, proof of Lemma 4.1] In the proof of Theorem 4.3 the formula for Adj_n is written with denominator (n−2)!; Lemma 4.1 with k = 3 gives (n−3)!. This is a typo in the displayed computation but should be corrected to avoid confusion.
[Throughout] There are numerous typographical errors: 'summmands', 'intruduce', 'decompoisition', 'distungiuish', 'accalerate', 'a a usage'. These should be cleaned up in a revision.
[Section 1, Remark 1.1] The notation Δ1 is used both for the full first cohomological Laplacian and, in the discussion of the Ozawa expression, for its components; a short notational reminder when switching between Δ1 and Δ±1 would improve readability.
[Remark 5.3] Remark 5.3 says the computations were done 'over a subspace of the group ring RG_n' and then claims this implies the H_n statement because H_n is a quotient. The intended meaning is clear, but the wording should say the computation was done in the quotient or after verifying invariance, otherwise the implication is not immediate.
[References] Reference [BS23] is cited as an arXiv preprint; if a published version is now available, it should be updated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: base cases are independent computer-assisted certificates and the induction is proved in-text; self-citations are technical, not conclusion-bearing.

full rationale

The paper's induction argument is self-contained. Lemma 4.1 and Lemma 4.2 are proved in the text by orbit�stabilizer and counting arguments, and Theorem 4.3 algebraically combines them to push an Adj-block sum-of-squares certificate from rank m to rank n. The base cases in Lemma 5.2 are independent computer-assisted certificates for H3, obtained with interval arithmetic and Wedderburn decomposition, not restatements of the target inequality. The use of [Miz24] and [KMN25] supplies the induction template and the certification method; these are prior technique results, not assumptions equivalent to the conclusion. No displayed equation defines Adj, Sq, Mono, or Op in terms of the desired inequality, and no parameter fitted to a subset is renamed as a prediction. I therefore find no circular step. Separately, Theorem 0.2 in the introduction states a capital-Delta_1 conclusion while Theorem 4.3 proves the Adj-block version; this is a statement/proof mismatch and a correctness risk, not a circularity.

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

The paper does not introduce new mathematical objects. It relies on external theorems for the sum-of-squares/property (T) equivalence, on a known presentation of Sp_{2n}(Z), and on the correctness of the numerical certification pipeline. No free parameters are fit to external data; the constants 0.24, 0.99, and 0.82 are outputs of the SDP computations, not adjustable inputs.

assumptions (4)

standard math The Bader-Nowak and Bader-Sauer results equating a sum-of-squares decomposition of Δ_1 - λI with property (T) for finitely presented groups are valid.
Used throughout as the framework for defining and interpreting spectral gaps; cited from [BN20] and [BS23].
domain assumption The Steinberg presentation of Sp_{2n}(Z) with the listed generators and relators is correct.
Section 2 introduces the presentation following [DK23], [Mat69], and [Beh75]; all computations rely on this presentation.
domain assumption The numerical SDP solutions obtained with SCS/JuMP can be rigorously certified using the interval arithmetic method of [KMN25] and the improved certification in Appendix B.
Lemma 5.2 and Theorem 5.6 depend on turning floating-point SDP output into exact sum-of-squares certificates; the soundness of this pipeline is assumed.
domain assumption The Groups.jl package correctly implements the symplectic group Sp_{2n} and its relations.
Remark 5.3 states computations were done over a subspace of RG_n using Groups.jl; a bug in the package would invalidate the base-case certificates.

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Pith. "Pith review of Inducing spectral gaps for the cohomological Laplacians of $\operatorname{Sp}_{2n}(\mathbb{Z})$." pith.science (2026). https://pith.science/paper/AF4XPURM

@misc{pith2026250416625,
  author       = {Pith},
  title        = {Pith review of: Inducing spectral gaps for the cohomological Laplacians of $\operatornameSp_2n(\mathbbZ)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AF4XPURM}},
  note         = {Machine review of arXiv:2504.16625}
}

abstract

We show that the spectral gap of the first cohomological Laplacian $\Delta_1$ for $\operatorname{Sp}_{2n}(\mathbb{Z})$ follows once a slightly stronger assumption holds for some $\operatorname{Sp}_{2m}(\mathbb{Z})$, where $n\geq m$. As an application of this result, we provide explicit lower bounds for some quotients of $\operatorname{Sp}_{2n}(\mathbb{Z})$ for any $n\geq 2$.

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Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

asentation von C hevalley-gruppen \

Helmut Behr. Explizite P r\"asentation von C hevalley-gruppen \"uber Z . Math. Z. , 141:235--241, 1975

work page 1975

[2] [2]

Julia: A fresh approach to numerical computing

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2017

[3] [3]

Uri Bader and Piotr W. Nowak. Group algebra criteria for vanishing of cohomology. J. Funct. Anal. , 279(11):108730, 18, 2020

2020

[4] [4]

Higher kazhdan property and unitary cohomology of arithmetic groups

Uri Bader and Roman Sauer. Higher kazhdan property and unitary cohomology of arithmetic groups. https://arxiv.org/abs/2308.06517, 2023

arXiv 2023

[5] [5]

Braid groups and symplectic S teinberg groups

Fran cois Digne and Christian Kassel. Braid groups and symplectic S teinberg groups. Ann. K-Theory , 8(4):669--692, 2023

work page 2023

[6] [6]

Ralph H. Fox. Free differential calculus. I . D erivation in the free group ring. Ann. of Math. (2) , 57:547--560, 1953

1953

[7] [7]

Ralph H. Fox. Free differential calculus. II . T he isomorphism problem of groups. Ann. of Math. (2) , 59:196--210, 1954

work page 1954

[8] [8]

G. D. James. The representation theory of the symmetric groups , volume 682 of Lecture Notes in Mathematics . Springer, Berlin, 1978

1978