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This paper proves that a random hyperbolic surface in the Brooks–Makover model has first Laplacian eigenvalue λ1 > 1/4 − ε with probability converging to 1, confirming the nearly optimal spectral gap conjecture for this model.

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2026-08-04 00:09 UTC pith:BKUS3MYT

load-bearing objection Credible proof of the last open random-surface spectral-gap conjecture, with one load-bearing enumeration gap that should be fixed before the result is regarded as settled. the 3 major comments →

arxiv 2511.02517 v3 pith:BKUS3MYT submitted 2025-11-04 math.SP math.DGmath.GTmath.RT

Nearly optimal spectral gaps for random Belyi surfaces

Yang Shen , Yunhui Wu This is my paper
classification math.SP math.DGmath.GTmath.RT MSC 58J5030F1005C80
keywords random Belyi surfacesBrooks-Makover modelspectral gapfirst Laplacian eigenvaluehyperbolic surfacesstrong convergencerandom permutationsPSL(2,Z)
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Random closed hyperbolic surfaces of large genus cannot have first Laplacian eigenvalue above 1/4, and 1/4 is conjecturally achievable. This paper proves that in the Brooks–Makover model—random surfaces glued from 2n ideal triangles according to a random 3-regular graph with orientation—the first eigenvalue exceeds 1/4 − ε for every ε > 0, with probability tending to 1. It is the last of the three standard models of random closed hyperbolic surfaces for which this nearly optimal spectral gap conjecture was open, so the result completes a common picture. The proof works by reinterpreting each Brooks–Makover surface as a degree-6n cover of the modular surface H/PSL(2,Z) and proving strong convergence of the associated random permutation representations of PSL(2,Z); since the modular surface itself has λ1 > 1/4, the gap transfers to the random covers.

Core claim

For any ε > 0, with probability tending to 1 as n → ∞, the closed conformal compactification S_C(Γ,O) of a random Brooks–Makover cusped surface S_O(Γ,O) satisfies λ1(S_C) > 1/4 − ε. Equivalently, the random cusped surface S_O has no eigenvalues in (0, 1/4 − ε). The paper establishes this by giving a new geometric description of the Brooks–Makover ensemble: each surface is isometric to a degree-6n covering of H/PSL(2,Z) coming from a pair (σ,τ) of permutations, σ of order 2 and τ of order 3, and then proving that the permutation representation of PSL(2,Z) defined by (σ,τ) is strongly convergent to the regular representation as n grows. The large-cusps lemma transfers the spectral gap from the

What carries the argument

The central device is the identification of a Brooks–Makover surface with the covering surface PSL(2,Z) \(Φ H × [6n]) determined by the homomorphism Φ: PSL(2,Z) → S_{6n} sending the standard generators to σ,τ. The proof then follows the polynomial method for strong convergence: expected traces of random permutation matrices are expanded in powers of 1/(6n), with coefficients controlled by counting fixed points of random permutations. Fixed points are counted by X-labeled graph homomorphisms from the word graph Γ_X(ω) to the permutation graph Γ(σ,τ): surjective homomorphisms are enumerated by a recursive algorithm (Operations I/II) and injective ones by falling-factorial expectations; the lea

Load-bearing premise

The load-bearing assumption is the exhaustiveness of the recursive algorithm in Section 3.2.2—that it generates every equivalence class of surjective X-labeled graph homomorphisms, so the bounds on |F_h(ω)| and the fixed-point expectation are complete—which the paper asserts but does not prove by an explicit bijection.

What would settle it

Compute, for a small word ω (such as ω = x1 x2 x1 x2^2 x1 x2 x1 x2 x1 x2^2 used as an example in the paper), all surjective X-labeled graph homomorphisms from Γ_X(ω) to subgraphs of Γ(σ,τ) for small n, and verify whether the recursive Operations I/II produce every equivalence class. Exhibiting one missed class, or showing |F_h(ω)| > k^{2h} for some h, would invalidate Lemma 3.5 and the expected-trace expansion.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For every ε > 0, with probability tending to 1, the closed Brooks–Makover surface S_C(Γ,O) has λ1 > 1/4 − ε; the same holds for the cusped surface S_O.
  • The Brooks–Makover model joins the other two standard models in which the nearly optimal spectral gap conjecture is now known to hold.
  • The strong-convergence result for permutation representations of PSL(2,Z) applies to every finite-support element of the group algebra, so traces and norms of arbitrary words converge to their regular-representation limits.
  • Because the base modular surface has λ1 > 1/4, the proof transfers a spectral gap from a fixed base to random covers, the same mechanism used in covering models.
  • The abstract's quantitative version (1/4 − c/log n for a universal c > 0) is stronger than the ε statement, giving an explicit rate of convergence in addition to the asymptotic result.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extension: If the explicit constants in the trace expansion are tracked, the method may yield effective finite-n lower bounds for λ1, not just an asymptotic statement.
  • Extension: The description of the Brooks–Makover ensemble as an asymptotically measure-zero subset of all PSL(2,Z)-homomorphisms suggests that other geometric statistics—diameter, systole, Cheeger constant—could be attacked with the same permutation-model machinery.
  • Extension: The counting algorithm might generalize from PSL(2,Z) ≅ Z2 ⋆ Z3 to other free products of finite cyclic groups, giving spectral-gap results for random surfaces built from ideal polygons with more sides.
  • Extension: The divisor-counting function d(µ) appearing in the leading coefficient is the same arithmetic input seen in random-cover eigenvalue statistics, hinting that the method could be pushed toward the question of a positive proportion of Ramanujan-like surfaces with λ1 ≥ 1/4.

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

3 major / 4 minor

Summary. The paper proves that a random hyperbolic surface in the Brooks–Makover model has first Laplacian eigenvalue larger than 1/4 - ε with probability tending to 1, for every ε > 0. The proof gives a new description of the Brooks–Makover ensemble as degree-6n covers of the modular surface with constrained generators (σ of order 2 with no fixed points, τ of order 3 with no fixed points), preserving the uniform measure up to a constant fiber size (Prop. 2.4). The main technical work is a fixed-point count for words in F_2: fixed points of the random permutation are encoded as X-labeled graph homomorphisms into the associated graph Γ(σ,τ) (Lemma 3.1), and the homomorphism set is decomposed into surjective and injective parts. Proposition 3.7 gives an asymptotic expansion for E_n[fix_ω], Proposition 4.1 converts this into a trace expansion for the permutation representation of PSL(2,Z), and Proposition 4.3 gives the needed norm bound u1(α^p). Following the template of Magee–Puder–van Handel, these imply strong convergence (Thm. 4.6), spectrum coincidence for the cusped surfaces (Cor. 4.7), and finally Theorem 1.1 via the large-cusps comparison of Brooks–Makover.

Significance. If correct, this is a breakthrough: it completes the nearly optimal spectral gap conjecture for all three principal models of random closed hyperbolic surfaces. The paper contains genuinely novel ingredients: a measure-preserving reformulation of the Brooks–Makover model as a constrained random PSL(2,Z)-cover, and a graph-theoretic enumeration scheme for the fixed-point counts with explicit, non-fitted coefficients u_i(γ). The announced theorem is an unambiguous, falsifiable asymptotic statement, and the main line of argument — trace expansion plus norm bound implying strong convergence — is the now-standard and credible polynomial-method template. However, the paper's central combinatorial enumeration step contains an asserted-but-unproven completeness statement, and the norm-bound proof uses an unexplained positivity reduction; these are load-bearing and must be repaired before the result can be considered established.

major comments (3)
  1. [§3.2.2, Lemma 3.5 and Proposition 3.7] The algorithm on pages 24–26 is introduced as one that will 'output all possible elements in I(ω)', but no formal proof of completeness is given. Proposition 3.7 uses the equality #{(g,Γ)∈I(ω): η(Γ)=h} = |F_{h+1}(ω)| and the bound |F_h(ω)| ≤ k^{2h} to control the infinite sum over I(ω). If any equivalence class of surjective homomorphisms is not generated by the recursive choices, or if the parameter a is not equal to η(G)+1 for every generated class, then the asymptotic expansion of E_n[fix_ω] is unsupported, and Propositions 4.1, 4.3 and Theorem 4.6 collapse. The manuscript needs an explicit bijection or induction showing that every element of I(ω) is realized by exactly one run of the algorithm up to the defined equivalence, with the bookkeeping invariant a=η+1 verified for all classes, not only for outputs of the algorithm.
  2. [§3.2.2, Lemma 3.6 and Proposition 3.8] Even if the algorithm is complete, Proposition 3.8's identity v_0(ω)=|F_1(ω)|=d(µ) requires more than Lemma 3.6 supplies. Lemma 3.6 identifies the target graph G as Γ_X(ν) with ω=ν^q, but it does not show that for a fixed divisor q there is exactly one equivalence class of surjective homomorphisms f: Γ_X(ω) → Γ_X(ν). The lower bound constructs one natural homomorphism per divisor, but the upper bound |F_1(ω)|≤d(µ) needs a uniqueness argument. Without it, v_0(ω), and hence u_1(γ) in Lemma 4.2, could be larger than d(µ)-1, affecting the norm bound in Proposition 4.3.
  3. [§4.3, proof of Proposition 4.3] The proof states: 'From [MdlS24, Proposition 6.3], we can assume that α has positive coefficients' and then normalizes Σ a_γ=1. This reduction is not immediate, because u_1 is linear on C[PSL(2,Z)] while the operator norm ∥λ(α)∥ is not linear in α. For a general self-adjoint α with signed coefficients, replacing α by a positive-coefficient version changes the value of ∥λ(α)∥, so the inequality for the modified polynomial does not imply the required inequality for the original α. The authors must either state the precise form of [MdlS24, Prop. 6.3] that applies to this u_1 and to the signed expansion u_1(α^p), or prove Proposition 4.3 directly for signed α (e.g. by decomposing α into positive and negative parts and using the rapid decay property invoked earlier in the same paragraph). As written, this is a load-bearing gap in the proof of strong convergence.
minor comments (4)
  1. [§2.2, Lemma 2.2] Typo: 'there there exists' should be 'there exists'.
  2. [§1] The name Bollobás is spelled 'Bollabás' in the introduction; also the notation F⋆_n is used inconsistently in a few places (e.g. the proof of Theorem 4.8 writes (σ,τ)∈F⋆_n where (Γ,O)∈F⋆_n is meant).
  3. [§3.2.1, proof of Proposition 3.2] The proof refers to 'Lemma 3.2' while the statement is Proposition 3.2; the numbering should be corrected.
  4. [§4.4, Lemma 4.4] The proof of part (2) says 'the proofs for the cases of c and c^2 are similar, that we leave to interested readers.' Since Lemma 4.4 is used essentially in Proposition 4.3, the omitted cases should be written out or at least summarized in enough detail for verification.

Circularity Check

0 steps flagged

No significant circularity: the spectral gap result is derived from independent fixed-point enumeration and external theorems; the target eigenvalue is not used as input.

full rationale

The paper's derivation chain is self-contained relative to its external inputs. Proposition 4.1 constructs the coefficients u_i(γ) from the fixed-point expectation E_n[fix_ω] via Propositions 3.7 and 3.8, where u_1(γ) is computed combinatorially as d(µ)−1 from primitive-word divisors; no spectral quantity (and in particular no value of λ_1) is fed into the expansion. Proposition 4.3 proves a genuine norm bound for this derived u_1 using Lemma 4.4, the rapid decay property, and the Cauchy–Schwarz estimates of Lemma 4.5; it does not assume the desired strong convergence. The proof of Theorem 4.6 then follows the external framework of [MPvH25, HM23, MdlS24], and the final spectral conclusion uses λ_1(H/PSL(2,Z)) > 1/4 from [Ber11, Theorem 3.38] plus the Brooks–Makover large-cusps results [BM04, BM01]. The only notable gap flagged by a careful reading is the asserted completeness of the §3.2.2 enumeration algorithm ('the algorithm... output all possible elements in I(ω)'), which is a correctness/completeness concern rather than a circularity: the counted graph homomorphism classes are not fitted to, nor defined in terms of, the target spectral gap. The authors' self-citations [SW23, SW25] appear only as contextual references and are not load-bearing. Accordingly no circular step satisfying the evidentiary standard is present.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The central claim rests on the strong-convergence-to-spectral-gap machinery (MPvH25/HM23), the large-cusps transfer (BM01/BM04), the spectral gap of the modular surface, and the new counting engine of Section 3. No numerical free parameters are fitted; the universal constants C, C_0, etc. are proof devices, not fitted values. No new physical/mathematical entities are postulated.

axioms (7)
  • domain assumption No spectrum of the modular surface H/PSL(2,Z) below 1/4−ε (stated as λ1(H/PSL(2,Z)) > 1/4).
    Invoked in Theorem 4.8 to conclude random covers have no small eigenvalues; cited from [Ber11, Theorem 3.38]. As written, the modular surface has continuous spectrum starting at 1/4, so '> 1/4' is convention-dependent; the needed fact is absence of spectrum below 1/4−ε.
  • domain assumption Brooks–Makover large cusps property: random S_O(Γ,O) has cusps of length ≥ l with probability → 1.
    Proposition 2.1, cited from [BM04, Theorem 2.1A]; needed to transfer spectral gap from cusped to compactified surface via Lemma 2.2.
  • domain assumption Transfer lemma λ1(S_C) ≥ min(1/4−ε, C(l,ε)λ1(S_O)) for surfaces with large cusps.
    Lemma 2.2, cited from [BM01, Lemma 1.2]; used in the final step of Theorem 4.8.
  • domain assumption Strong convergence of random representations implies spectral coincidence below 1/4−ε (Hide–Magee argument).
    Theorem 4.8 uses 'following [HM23]' to convert Cor 4.7 into Spec(Δ_{H/PSL(2,Z)})∩(0,1/4−ε) = Spec(Δ_{S_O})∩(0,1/4−ε).
  • standard math PSL(2,Z) ≅ Z_2 ⋆ Z_3 with generators b of order 2 and c of order 3.
    Used throughout Section 4 to parameterize Φ(σ,τ) and to reduce trace computations to word counting.
  • standard math PSL(2,Z) has rapid decay; C*_red[PSL(2,Z)] is simple; PSL(2,Z) has the unique trace property.
    Used in Theorem 4.6 lower bound and Proposition 4.3; cited from [Jol90, BCdlH94, BKKO17] and [dlH00].
  • standard math Fixed points of random permutation words correspond to X-labeled graph homomorphisms, per the Puder–Parzanchevski framework.
    Lemma 3.1 establishes the correspondence; the framework is cited from [PP15].

reviewed 2026-08-04 · how reviews work

0 comments
read the original abstract

In this paper, we show that a random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than $\left(\frac{1}{4}-\frac{c}{\log n}\right)$ for some universal constant $c>0$ , confirming the nearly optimal spectral gap conjecture in this model.

Figures

Figures reproduced from arXiv: 2511.02517 by Yang Shen, Yunhui Wu.

Figure 1
Figure 1. Figure 1: The standard hyperbolic ideal triangle T [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Orientation-preserving Definition. The compact Riemann surface S C (Γ, O) is defined as the conformal compactification of S O(Γ, O) by filling in all the punctures. For any fixed n ∈ N, F ⋆ n is finite, hence there is a classic probability measure Probn BM on it. For any random variable f : F ⋆ n → R, let E n BM[f] be its expected value. More precisely, Probn BM(A) = |A| |F⋆ n | and E n BM[f] = P (Γ,O)∈F⋆ … view at source ↗
Figure 3
Figure 3. Figure 3: Tilings Fγ,i def = {(γp, i); p ∈ F}, which is a subset of H × [6n], and write Fi = Fe,i for simplicity, where e is the unitary element in PSL(2, Z). Then we have (i) for any γ ∈ PSL(2,Z) and 1 ≤ i ≤ 6n, Fγ,i is identical to FΦ(γ−1)(i) in S; (ii) for any 1 ≤ i ̸= j ≤ 6n, Fi and Fj have no interior intersection in S. Hence, the quotient space S could be obtained by gluing all F ′ i s along their sides. For a… view at source ↗
Figure 4
Figure 4. Figure 4: Two examples for S O(σ, τ ) (i) for 1 ≤ t ≤ 2n, as the second one of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two X−labeled graphs By construction, it is easy to see that for any (σ, τ ) ∈ En (n ≥ 1), the X−labeled graph Γ(σ, τ ) satisfies (i) the vertex set V (Γ(σ, τ )) = {v1, ..., v6n}, and for each vertex vi (1 ≤ i ≤ 6n), there are exactly four different edges based at it: two outgoing edges x1, x2 and two incoming edges x1, x2; (ii) the edge set E(Γ(σ, τ )) consists of cycles that are of the following two form… view at source ↗
Figure 6
Figure 6. Figure 6: Completion of ΓX(ω) For any two elements (g1, Γ1), (g2, Γ2) ∈ F(ω), we say that (g1, Γ1) and (g2, Γ2) are equivalent if there exists an X−labeled graph isomorphism ξ : Γ1 → Γ2 such that g2 = ξ ◦ g1, i.e. the following diagram commutes: Γ1 ΓX(ω) Γ2 ξ g1 g2 . Denote by I(ω) = the set of all equivalent classes in F(ω). For any X−labeled graph homomorphism f : ΓX(ω) → Γ(σ, τ ), let Γ be the image f Ä ΓX(ω) ä ⊂… view at source ↗
Figure 7
Figure 7. Figure 7: Two cycles Recall that the word ω ∈ F2 is of form ω = x1x i1 2 · · · x1x ik 2 where k ≥ 1, ij ∈ {1, 2} (1 ≤ j ≤ k). For any element (g, Γ) ∈ I(ω), since g maps xi−cycles in ΓX(ω) to xi−cycles in Γ(σ, τ ) for i ∈ {1, 2}, it follows that Γ = g Ä ΓX(ω) ä is a union of certain x1−cycles and x2−cycles. Moreover, it satisfies that (i) for any vertex v ∈ V (Γ), it is contained in some an x2−cycle of Γ; (ii) for a… view at source ↗
Figure 8
Figure 8. Figure 8: Relabel ΓX(ω) when ω = x1x 2 2x1x2x1x2 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: An output for ω = x1x2x1x 2 2x1x2x1x2x1x 2 2 Example. Now we consider an example for ω = x1x2x1x 2 2x1x2x1x2x1x 2 2 containing all possible operations above (see [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.