Limit points of uniform arithmetic bass notes

Bram Petri; Will Hide

REVIEW 4 major objections 5 minor 2 cited by

Limit points of uniform arithmetic bass notes

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The set of limit points of spectral gaps of closed arithmetic hyperbolic surfaces is exactly the full interval $[0,\frac14]$.

desk verdict Answers Magee's question with a genuine new construction; the proof is coherent and the computation is reproducible, but the key convergence theorem rests on unpublished work. read the letter →

arxiv 2412.15111 v1 pith:PDZNLWOS submitted 2024-12-19 math.SP math.DGmath.GTmath.NTmath.PR

Will Hide , Bram Petri This is my paper · ORCID

classification math.SPmath.DGmath.GTmath.NTmath.PR MSC 58J5011F72

keywords spectralgaparithmetichyperbolicsurfaceslimitpointsLaplacianeigenvaluesrandomcoversBolzasurfacestrongconvergencedegree-two

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

This paper answers a question left open by the recent non-compact case: for closed arithmetic hyperbolic surfaces, the possible spectral gaps are dense in the interval $[0,\frac14]$. The main theorem states that the values $\lambda_1(X)$ attained by closed arithmetic surfaces, intersected with $[0,\frac14]$, form a dense subset of $[0,\frac14]$. Since there are only finitely many arithmetic surfaces of any fixed genus, this is the same as saying that the set of limit points of all such spectral gaps is exactly $[0,\frac14]$. The proof builds, inside one fixed arithmetic family (covers of the Bolza surface), finite covers whose spectral gaps approximate every number in the interval. This matters because it turns a countable, seemingly sparse set into a continuum, showing that arithmeticity alone imposes no further restriction on possible gaps in this range.

What carries the argument

The proof is carried by a random-model construction. Fix a handlebody attachment on the Bolza surface (the genus-two surface obtained from a regular octagon with opposite sides identified) and a pants decomposition by meridians; a homeomorphism $\phi$ produces an epimorphism from the surface group $\Gamma_B$ to a free group $F_2$, whose kernel gives an infinite planar 'tree cover' $T_\phi$. Choosing a uniformly random permutation representation $\rho\in\mathrm{Hom}(F_2,S_n)$ yields a random finite cover $Y_\phi^{(n)}\to X_B$, and homomorphisms of $\pi_1(Y_\phi^{(n)})$ to $\mathbb{Z}/2\mathbb{Z}$ give the degree-two covers whose spectral gaps are studied. Three ingredients make the argument work: random pants-decomposition asymptotics select $\phi$ so that $T_\phi$ has long systole, wide standard collars, and spectral gap close to $\frac14$; a strong-convergence theorem for permutation representations transfers this gap to the random covers; and a 'switching' move changes one degree-two cover into another along a preferred pants curve without moving the spectral gap by more than a prescribed $\eta$, using eigenfunction delocalization and flattening near collars. A separate computer-assisted linear-programming check based on the Selberg trace formula certifies that every connected degree-two cover of the Bolza surface has $\lambda_1>\frac14$, which is what makes the near-$\frac14$ double covers possible.

What would settle it

Find a uniform lattice $\Gamma$, an epimorphism $\Gamma\to\Lambda$, and a sequence of permutation representations of $\Lambda$ whose standard representations converge strongly to the regular representation, yet for which the smallest positive eigenvalues of the resulting covers $\Gamma_n\backslash\mathbb{H}^2$ do not converge to $\min\{\lambda_1(K\backslash\mathbb{H}^2),\lambda_1(\Gamma\backslash\mathbb{H}^2)\}$; such an example would invalidate Theorem 6.2 and break the construction of covers with gaps near $\frac14$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: the set $\{\lambda_1(X): X \text{ a closed arithmetic surface}\}\cap[0,\frac14]$ is dense in $[0,\frac14]$. Equivalently, as stated in the abstract, the limit points of the spectral gaps of closed arithmetic hyperbolic surfaces are precisely the whole interval $[0,\frac14]$. The surfaces that realize this density are built as degree-two covers of a certain sequence of random finite covers of the Bolza surface, so the density is achieved using only finite-index torsion-free subgroups of a single uniform arithmetic group, the $(2,3,8)$-triangle group (generated by rotations of orders 2, 3 and 8 fixing a hyperbolic triangle). The same method proves an analogous statement for any closed orientable hyperbolic surface $X=\Gamma\backslash\mathbb{H}^2$: the spectral gaps of its finite-index subgroups are dense in $[0,\Lambda]$, where $\Lambda$ is the minimum spectral gap among double covers of $X$. It also yields a new proof that some sequence of finite-sheeted covers of any such $X$ has genus tending to infinity and spectral gap tending to $\frac14$.

Load-bearing premise

The load-bearing premise is that the unpublished strong-convergence theorem cited as [Mag24b] is true as stated, with only a proof sketch supplied; a secondary gap is that the cycle-counting statement used in Section 6.6 is asserted to follow by similar methods but is not contained in any cited reference.

Editorial extensions

If this is right

Because the density is achieved by covers of the Bolza surface, the interval $[0,\frac14]$ is realized inside a single arithmetic commensurability class, not by combining many different classes.
For any closed orientable hyperbolic surface $X$, the spectral gaps of its finite-index covers are dense in $[0,\Lambda]$, where $\Lambda$ is the smallest spectral gap among its double covers.
There exists a sequence of finite-sheeted covers of any such $X$ with genera tending to infinity and spectral gaps tending to $\frac14$, recovering a previously known result by a different argument.
No value strictly above $\frac14$ can be a limit point of spectral gaps of closed arithmetic surfaces; values above the interval can occur only as isolated points, such as those coming from low-genus exceptional surfaces like the Bolza surface and the Klein quartic.

Reading between the lines

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

The authors do not spell this out, but the switching construction suggests a testable generalization: any base surface whose double covers all have spectral gap above $\frac14$ and which admits meridian pants decompositions with arbitrarily long curves should yield the same density statement for its finite covers.
If the unpublished strong-convergence ingredient is published in the stronger form sketched here, the same random model would likely give convergence not only of the first eigenvalue but of the full spectral measure of the random covers to that of the tree cover, making the density proof part of a finer spectral equidistribution statement.
The constants in the proof are uniform, so a quantitative version could in principle be extracted: for a given tolerance $\eta$, one could bound the cover degree needed to place a spectral gap within $\eta$ of any prescribed value in $[0,\frac14]$.

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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

This paper answers Magee's question: the spectral gaps of closed arithmetic hyperbolic surfaces are dense in [0,1/4]. That is the headline. The proof is a genuine new construction, not a cosmetic variant of the non-compact case. They build random covers of the Bolza surface via maps to F2, use Mirzakhani's pants-decomposition asymptotics to get a tree cover with long systole, wide collars, and λ1 close to 1/4, then interpolate to lower gaps by switching among degree-two covers. The computer check that every degree-two cover of Bolza has λ1 > 1/4 is shipped as a notebook, and the interval arithmetic looks reproducible. Credit where due: this is a serious piece of work.

The soft spots are the usual ones for this area. Theorem 6.2, the strong-convergence step that identifies the limit of λ1 of the random covers, is stated with a proof sketch and depends on Magee's unpublished [Mag24b]. The text says the precise statement is not yet in the literature. That is load-bearing: if [Mag24b] fails to appear or doesn't cover this setting, Theorem 6.4 doesn't go through. The reliance is at least explicit and the authors cite closely related published work (Hide–Magee, Louder–Magee, Magee–Thomas) that plausibly contains the needed ingredients. The Nica-type independence claim in Section 6.6 is also stated as not literally in the literature, but they list four references with versions of it; that's a minor gap, not a red flag. On the presentation side, the abstract says 'limit points equals [0,1/4]' while the body proves density. Density plus a standard upper bound λ1 ≤ 1/4 + O(1/g) gives the reverse inclusion, but the upper bound isn't stated, so the abstract does a bit more work than the text. That's a fixable presentation gap.

My bottom line: the central argument is coherent and the result is significant. I would send this to a serious referee. The referee should be asked to verify Theorem 6.2's provenance and whether the proof sketch can be made complete without [Mag24b]. If the authors can either prove Theorem 6.2 in the paper or point to a public version of Magee's manuscript, I'd be comfortable accepting after minor revisions. If [Mag24b] remains unavailable, this should be conditional pending that paper.

I'd bring this to our reading group and I'd cite it.

Referee Report

4 major / 5 minor

Summary. Hide and Petri study the set of first nonzero eigenvalues (spectral gaps) of closed arithmetic hyperbolic surfaces. Their main theorem asserts that this set is dense in [0, 1/4]. The strategy starts from the Bolza surface X_B, a fixed handlebody attachment, and a meridian pants decomposition, then forms random covers Y_φ^(n) whose monodromy factors through a free group F2 via a homeomorphism φ. A tree cover T_φ is constructed with large systole, wide collars, and λ1 close to 1/4, using Mirzakhani's distribution of random pants decompositions and Cheeger's inequality. Strong convergence of permutation representations (Bordenave–Collins) together with a spectral-convergence theorem for covers (Theorem 6.2) is used to obtain random covers and degree-two covers with gap close to 1/4. Intermediate gaps are reached by switching between degree-two covers: Proposition 5.3 bounds the change in the Rayleigh quotient under a switch, and a Nica-type cycle-count statement is used to ensure that the switched curves have wide collars and long systole with positive probability. A computer-assisted linear programming verification (Proposition 3.2) shows all degree-two covers of X_B have λ1 > 1/4, which is used to identify the limiting gap in the strong-convergence step. The paper also claims, in the abstract and introduction, that the set of limit points of all such spectral gaps equals [0, 1/4].

Significance. If the proof is completed, this is a substantial contribution: it answers Magee's question for closed arithmetic surfaces and provides a new proof of the Louder–Magee theorem on covers with near-optimal spectral gaps. The paper has clear strengths: the order of constants is explicit in Section 6.1; the computer verification is shipped as a Jupyter notebook using interval arithmetic; the geometric inputs are stated as precise propositions; and the overall architecture is coherent and well motivated. However, the central result is currently conditional on an unpublished theorem and on an unproven extension of Nica's cycle-count result, and the abstract overstates what is proved. These issues need to be resolved before the paper can be accepted.

major comments (4)

[Section 6.3 (Theorem 6.2)] Theorem 6.2 is the spectral-convergence engine: it is used in Theorem 6.4 to guarantee that λ1(Y_φ^(n)) and the chosen degree-two cover have λ1 > 1/4 − η. The proof is only a sketch and explicitly states that the precise statement is not yet available in the literature, citing the unpublished manuscript [Mag24b] for its two key steps (Theorem 3.1 and Proposition 2.9). If [Mag24b] is not available, or if its hypotheses do not match the present setting, the construction of good near-1/4 covers collapses. The paper must either include a complete proof of Theorem 6.2 or be revised to state the main theorem as conditional on a precisely cited published result.
[Section 6.6 (Nica-type cycle statistics)] The positive-probability control of short geodesics and collar widths relies on a vector-independence and Poisson-limit statement for cycle counts of random permutations under word maps. The text says this statement "is not literally contained in Nica's paper, but it can be proven using similar methods", referring to [LP10, BP23, PZ24] without giving a precise statement. This is load-bearing for the switching argument, specifically for avoiding a geodesic γ satisfying (6.1). Please provide a self-contained proof or a precise citation of a theorem whose hypotheses match the use.
[Abstract and Section 1.2] The abstract claims that the set of limit points of spectral gaps of closed arithmetic hyperbolic surfaces equals [0, 1/4]. What Theorem 1 actually proves is that the set of spectral gaps is dense in [0, 1/4]. Denseness implies that every point of [0, 1/4] is a limit point, but it does not rule out limit points larger than 1/4; indeed the Bolza surface itself has λ1 ≈ 3.8389. No argument is given excluding accumulation outside [0, 1/4]. Please either prove the reverse inclusion or rephrase the abstract and the introductory "can be stated as" sentence as a density result.
[Section 6.4 (proof of Theorem 6.4)] The existence of a degree-two cover of Y_φ^(n) with λ1 > 1/4 − η is obtained by asserting that θ ⊗ ρ_n strongly converges to ρ_reg "using the same argument as in [Mag24a, Section 4]". This is a second, unstated strong-convergence input with a quantitative conclusion that is essential to the construction. It should be stated as a lemma with either a proof or an exact reference rather than an appeal to an analogous argument elsewhere.

minor comments (5)

[Section 6.6] In the switch bound, the displayed inequality says ⟨∆f′, f′⟩/⟨f′, f′⟩ ≤ λ1( bY2) + η; from context it should be λ1( bY1) + η, since f is an eigenfunction on bY1 and f′ is constructed there. Please correct the label.
[Section 6.6] The phrase "asymptotically strictly positive" for the probability of the good event is imprecise; the proof gives a uniform lower bound a > 0 for the cycle-count event and a probability tending to 1 for the spectral event, which together imply a positive lower bound. Consider stating this directly.
[Global] There are several typos: "satsify" in Section 3, "homomoprhism" in Section 3, "desciption" in Section 6.5, and "Mirakhani" instead of "Mirzakhani" in the proof of Proposition 6.1.
[Figure 1] The caption "A tree cover of a surface of genus two" does not explain the relationship to the handlebody attachment or the meridian pants decomposition described in Sections 2.2–2.3; a more informative caption would help.
[Section 6.5] The identification of H^(n) with (Z/2Z)^k depends on a choice of spanning tree in W^(n), making the Hamming distance and the switch operation generating-set dependent. It may be worth noting explicitly that any two elements of H_φ^(n) are joined by switches in this generating set, which is what the connectivity argument needs.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is self-contained; the abstract's equality-of-limit-points wording is a presentation gap, not a circular step.

full rationale

The derivation chain is not circular. The paper fixes the target tolerance η and the auxiliary constants w and ℓ in Section 6.1 before any spectral-gap construction, and neither the target interval [0,1/4] nor the theorem being proved is used to choose these parameters. The high-gap tree cover T_ϕ is produced from Mirzakhani's random pants-decomposition asymptotics together with a Cheeger-type lower bound (Lemma 4.1, Proposition 4.2, Proposition 6.1); it is not obtained by assuming the desired spectral gap. The degree-two covers are handled by a switching estimate (Proposition 5.3) whose hypotheses—wide standard collars and long systole—are established with positive probability using Nica's cycle-counting results, again independently of the target density statement. The key spectral-convergence input, Theorem 6.2, is external to the construction: it is quoted as a version of results in the unpublished notes [Mag24b] and in prior strong-convergence work, and while its proof is only sketched and depends on material not yet in the literature—a genuine correctness fragility—it is neither defined in terms of nor derived from the target claim. The computer-assisted bound λ1(Y17) ≥ 0.2501 (Proposition 3.2) is an independent verification against an external trace-formula criterion, not a fit of the desired answer. There are self-citations ([HM23], [LM23], [FBP23], [FBGMPP23]), but they supply published techniques and benchmarks, not the conclusion of density in [0,1/4]. The abstract asserts that the set of limit points equals [0,1/4], while Theorem 1 only proves density in [0,1/4]; this is a presentation/correctness gap, not circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The proof relies on several external results, most of which are standard. The two load-bearing dependencies that are not fully established in the cited literature are Theorem 6.2 (which needs unpublished work of Magee) and the extended Nica-type permutation statistic. No free parameters are fitted to data; the constants η, w, ℓ, d are chosen by hand to satisfy explicit inequalities. No invented entities are introduced.

free parameters (5)

η (density parameter)
Chosen positive at the start; the goal is to build covers whose spectral gaps are η-dense in [0,1/4].
w (collar width)
Fixed independently of η; used in Proposition 5.3 and in the collar standardness requirements.
ℓ (systole threshold)
Chosen > 4w and satisfying explicit inequalities in Section 6.1 that link Cheeger bound, delocalization, and switch error.
d (trace formula test function parameter) = 3/4
Chosen in Proposition 3.2 so that supp(f_d) = [-3,3], allowing the finite computation; larger d would require more conjugacy classes.
0.2501 (eigenvalue check threshold) = 0.2501
A convenient lower bound above 1/4; the authors note no effort was made to optimize it.

assumptions (6)

ad hoc to paper Theorem 6.2: strong convergence of induced representations implies convergence of λ1 for covers of a uniform lattice
The precise statement is not available in the literature and the proof sketch depends on the unpublished [Mag24b]. Used to deduce λ1(Y^{(n)}) → min{λ1(T_φ), λ1(X_B)} and the existence of good 2-covers.
domain assumption Independence/cycle-count statement for random permutations following Nica [Nic94]
The paper states a version for arbitrary finite collections of non-trivial conjugacy classes, noting it is not literally in Nica's paper but provable by similar methods; used to ensure positive probability of systole and collar conditions.
domain assumption Mirzakhani's asymptotic of random pants decompositions [Mir16]
Used in Proposition 6.1 to find a homeomorphism φ producing a tree cover with long pants curves and near-optimal gap.
domain assumption The trace formula criterion of [FBGMPP23, Proposition 3.1]
Used in Proposition 3.2 to exclude small eigenvalues of Y_17; the criterion is taken from the authors' previous work.
standard math Bordenave-Collins strong convergence [BC19]
Random permutation representations of F2 converge strongly to the regular representation; used with Theorem 6.2.
standard math Bolza surface is arithmetic (Takeuchi [Tak77])
Ensures the constructed covers are arithmetic hyperbolic surfaces.

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Pith. "Pith review of Limit points of uniform arithmetic bass notes." pith.science (2026). https://pith.science/paper/PDZNLWOS

@misc{pith2026241215111,
  author       = {Pith},
  title        = {Pith review of: Limit points of uniform arithmetic bass notes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PDZNLWOS}},
  note         = {Machine review of arXiv:2412.15111}
}

abstract

We prove that the set of limit points of the set of all spectral gaps of closed arithmetic hyperbolic surfaces equals $[0,\frac{1}{4}]$.

Figures

Figures reproduced from arXiv: 2412.15111 by the authors.

**Figure 1.** A tree cover of a surface of genus two. 1. Introduction The spectral gap λ1(X) of a closed orientable hyperbolic surface X is the minimal non-zero eigenvalue of its Laplacian. A natural question, that fits into a larger context of the study of the bass note spectra of locally uniform geometries [Sar23], is which numbers can appear as spectral gaps of hyperbolic surfaces. It can be derived in various ways from variou… view at source ↗

**Figure 2.** A pair of pants β1 into two arcs, the shortest one of which has length at most 1+δ 2 L. If we suppose this shortest arc lies on the side of β2, we obtain ℓ(α) + 1 + δ 2 L ≥ ℓ(β2) ≥ L, which gives us ℓ(α) ≥ 1−ε 2 L and hence sys(Tϕ) ≥ (1 − δ) · L > ℓ. We prove item 2 using essentially the same argument. Indeed, if a collar of width u around one of the pants curves is not standard, then this yields a simple arc of len… view at source ↗

**Figure 3.** A local picture of a switch. The surface below represents a neighborhood of one of the prefered pants curves α (n) i of Y (n) ϕ . The surfaces up top (one on the left and one on the right) are the neighborhoods of the lifts of α (n) i in two distinct covers of degree two. The shading is there to indicate that the curves that are involved are meridians. To make this precise, we need to first decribe a free generating… view at source ↗

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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

Spectral gaps on thick part of moduli spaces
math.DG 2025-01 conditional novelty 7.0 of 10

For every fixed k and every ε>0, the maximum of λ_k - λ_{k-1} over genus-g hyperbolic surfaces with systole at least ε tends to 1/4 as g→∞.
Bass notes of random hyperbolic surfaces of large genus
math.SP 2026-07 accept novelty 2.0 of 10

A survey of recent results proving that random hyperbolic surfaces of large genus have near-optimal spectral gaps, after Hide–Magee, Anantharaman–Monk, and Hide–Macera–Thomas.

Reference graph

Works this paper leans on

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