Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups

Debattam Das; Krishnendu Gongopadhyay

REVIEW 3 major objections 4 minor 14 references

Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper derives an asymptotic formula for reciprocal conjugacy classes in Hecke groups using a central-limit count of reduced words.

desk verdict The CLT step in the main theorem is not merely underjustified; it makes the stated asymptotic false for r=10. read the letter →

arxiv 2506.08449 v1 pith:HKMC6E35 submitted 2025-06-10 math.GR math.GT

Debattam Das , Krishnendu Gongopadhyay This is my paper

classification math.GRmath.GT MSC 11F0620H0520H1020E45

keywords Heckegroupsreciprocalclassesreversibleelementsgeodesicswordlengthgrowthcentrallimittheoremfreeproducts

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 seeks to establish an asymptotic count of reciprocal conjugacy classes in the Hecke groups $\Gamma_p$ (the free products $\mathbb{Z}_2 * \mathbb{Z}_p$, including the modular group) as a function of word length. A reciprocal class is a conjugacy class containing an element conjugate to its own inverse; such classes correspond to infinite dihedral subgroups and to reciprocal geodesics on the associated hyperbolic orbifold. The main claim is that for odd $p=2r+1$ the count is asymptotic to one half of a sum of $(2r)^n$ terms, with the larger-$n$ part of the sum weighted by the standard normal cumulative distribution function with mean $r+3$ and variance $(r^2-1)/3$. For even $p=2r$ the paper proves a parallel estimate with base $(2r-1)^n$ and different mean and variance, and it shows that symmetric reciprocal classes dominate all reciprocal classes in that case. If correct, this gives the first word-length growth formula for reciprocal classes valid for arbitrary $p$, and it recovers the known modular-group count as the special case $r=1$.

What carries the argument

The machinery is a normal form for reciprocal words inside the free product $\mathbb{Z}_2 * \mathbb{Z}_p$. Lemma 2.8 says a cyclically reduced representative of a reciprocal class can be written as $$\iota\$gamma^{{k_1}}$\iota\$gamma^{{k_2}}$\cdots\iota\$gamma^{{k_n}}$\iota\$gamma^{{-k_n}}$\cdots\iota\$gamma^{{-k_2}}$\iota\$gamma^{{-k_1}}$,$$ so its word length is $\sum_{i=1}^n 2(|k_i|+1)$. Counting reciprocal classes of length at most $x$ therefore becomes a count of integer tuples $(k_1,\dots,k_n)$ satisfying $\sum_i 2(|k_i|+1) \le x$, summed over $n$, multiplied by the number of allowed $k_i$ choices, and divided by 2 because each class has two representatives of this form. The paper treats the block lengths $X_i = 2(|k_i|+1)$ as independent and identically distributed random variables, computes their mean and variance, and applies the Central Limit Theorem to replace, for each $n$, the exact proportion of tuples with total length below $x$ by the normal cumulative distribution function $\Phi((x-n\mu)/(\sqrt{n}\sigma))$. That replacement is the step that produces the explicit power-sum-plus-normal-tail formula.

What would settle it

Enumerate all reciprocal classes in a small Hecke group such as $\Gamma_5$ up to word length $x=200$ using the normal form, and compare the exact count with the right-hand side of Theorem 1.1; if the ratio does not approach 1 as $x$ grows, the asymptotic claim fails. A sharper test isolates the terms with $n$ near $\lfloor x/4\rfloor$, computing $\sum_n (2r)^n \mathbf{1}\{S_n \le x\}$ exactly and comparing it with the corresponding $\Phi$-weighted sum, since a non-vanishing relative error there would expose the failure of the normal approximation.

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Extended reading notes

Core claim

The central claim is Theorem 1.1. For odd $p=2r+1$, let $Q_x$ be the set of reciprocal classes in $\Gamma_p$ with word length at most $x$. The paper proves $$|Q_x| \sim \frac12 \sum_{n=1}^{\lfloor x/(2(r+1))\rfloor} (2r)^n + \frac12 \sum_{n=\lceil x/(2(r+1))\rceil}^{\lfloor x/4\rfloor} (2r)^n \Phi\!\left(\frac{x-n\mu}{\sqrt{n}\$\sigma$}\right),$$ with $\mu = r+3$ and $\sigma^2 = (r^2-1)/3$. For even $p=2r$, Theorem 1.2 says that the total number $|T_x|$ of reciprocal classes satisfies the same style of estimate with base $(2r-1)^n$, mean $\mu = (2r^2+4r-2)/(2r-1)$, and variance $\sigma^2 = (16r^3+36r^2+32r-12)/(6(2r-1))$, and with $\simeq$ in place of $\sim$ because the proof only establishes order equivalence. The paper also proves that primitive reciprocal classes are asymptotically all reciprocal classes, and that the modular-group case $p=3$ reduces to the known count $2^{\lfloor x/4\rfloor}$.

Load-bearing premise

The load-bearing premise is that the normal approximation to the distribution of block-length sums is uniformly accurate across every $n$ in the summation range, including the uppermost terms where the normal tail is extremely small; if the approximation degrades there, the reported ratio need not tend to 1.

Editorial extensions

If this is right

The dominant term in the odd case is the power sum with base $2r$, so the exponential growth rate of reciprocal classes is set by the size of the $\mathbb{Z}_p$ factor in the free product.
Because non-primitive reciprocal classes are negligible, almost every reciprocal class at large word length is primitive, and the same asymptotics describe primitive reciprocal geodesics.
For even $p$, the symmetric reciprocal classes alone determine the growth rate of all reciprocal classes; the $p$-reciprocal and mixed classes contribute only lower-order terms.
For $p=3$, the normal-tail part vanishes and the formula reduces to $\frac12(2^{\lfloor x/4\rfloor}-1)$, recovering the known modular-group count from a new word-length perspective.
Reading reciprocal classes as infinite dihedral subgroups, the result counts conjugacy classes of such subgroups in $\Gamma_p$ up to word length $x$.

Reading between the lines

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

The same normal-form-plus-CLT scheme should transfer to any free product $\mathbb{Z}_2 * \mathbb{Z}_k$ and to other triangle groups whose reciprocal words have a palindrome normal form; only the block-length distribution and the number of representatives per class would change.
A local limit theorem would refine the cumulative formula into an estimate for reciprocal classes of length exactly $x$, giving the same base with a Gaussian profile; the paper stops at the cumulative level.
The normal-tail sum could presumably be converted by a saddle-point estimate into a closed form of the type $C x^{-3/2} (2r)^{x/\mu}$ times a bounded oscillating factor, the standard shape for conjugacy-class growth in free products, which the paper leaves implicit.
For small $r$, the formula can be checked numerically against exact enumeration of reciprocal words, and the boundary between agreement and disagreement would map the range of word lengths where the Gaussian tail approximation is reliable.

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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

Dear Colleague,

Read the Hecke reciprocal-class paper. The idea is to use the CLT to replace exact counts of n-tuples with a normal tail, then sum over n. That is a legitimate move for the bulk of the sum, but the paper applies it in the large-deviation regime, where the normal tail overestimates the true probability. The result is that Theorem 1.1 is not left unproved; it is wrong.

What is good: the paper sets out a clean program, states explicit formulas for all Hecke groups, and recovers the modular group result as a corollary. The normal form in Lemma 2.8 (from the authors' earlier work) is useful. The idea of using probability to handle cumulative sums is worth exploring.

Where it breaks: Lemmas 3.2 and 3.3 replace P(S_n ≤ x) by Φ((x−nμ)/(√n σ)) for every n in the range, then multiply by (2r)^n and sum. For n near the middle of the range, (x−nμ)/(√n σ) → −∞, so the normal tail has a relative error that blows up; Berry–Esseen only gives absolute error O(1/√n). More decisively, the exact count of reciprocal words is a regular-language count with exponential rate ρ^{-1}, where ρ is the smallest positive root of 2Σ_{j=1}^{r} ρ^{2j+2}=1. For r=10, that rate is about 1.413, while the Gaussian sum in Theorem 1.1 has rate about 1.483, attained at n ≈ 0.187x. Two different exponential rates cannot be asymptotically equivalent. So the theorem is false as stated, not just poorly justified.

There are smaller problems: the proof of Theorem 1.2 uses a handwavy comparison |T_x| ≃ |Sym_x|, and the paper leans heavily on the authors' own [DG1] and [DG2] without independent verification. The modular-group corollary has a typo in the range (x/(r+1) should be x/(2(r+1))), but the result is correct.

Who is this for? A reader interested in a cautionary example of CLT misuse in group theory might find it useful. But as a research paper, it needs a fundamental repair before it can be taken seriously.

I would send it to a referee if it crossed my desk, because the question is real and the error is instructive, but I would not accept it as written.

Best, [You]

Referee Report

3 major / 4 minor

Summary. The paper studies reciprocal conjugacy classes in the Hecke groups Γ_p = Z/2 * Z/p, counting classes whose word length is at most x. Theorem 1.1 claims an asymptotic formula for odd p=2r+1 involving a finite geometric sum plus a sum of (2r)^n Φ((x-nμ)/(√nσ)) terms, and Theorem 1.2 claims an analogous '≃' formula for even p. The proofs use reciprocal normal forms taken from the authors' previous papers [DG1], [DG2], then replace the cumulative count of n-block words by a CLT approximation for the block-length distribution. Corollaries on primitive reciprocal classes and on the modular group are derived from the two theorems.

Significance. The probabilistic approach is a genuinely different route from the thermodynamic and combinatorial treatments in the literature, and the paper correctly identifies the normal forms and the mean/variance of the block lengths. However, the central asymptotic claim is not merely unproved: an exact generating-function count shows that Theorem 1.1 is false for r=10, and the same CLT replacement invalidates Lemma 3.2, Lemma 3.3, and the proof of Theorem 1.2. No machine-checked proofs, code, or independent verification of the key estimates are supplied, so the positive contribution is limited to the setup and the statement of a plausible but false conjecture.

major comments (3)

[Section 3, Lemma 3.2 and Lemma 3.3] The proof assumes that for every n in the summation range one has |P(S_n≤x)-Φ((x-nμ)/(√nσ))| < ε/2 with ε = (1/x)Φ((x-xμ)/(√xσ)). This cannot hold for the n near the upper endpoint: there z=(x-nμ)/(√nσ) ~ -c√x, so Φ(z) is exponentially small in x, whereas Berry-Esseen gives only a uniform absolute error O(1/√n)=O(1/√x), which is far larger than ε for large x. The inequalities in the displayed derivation after Eq. (3.2), and the identical step in Lemma 3.3, therefore do not follow from the CLT. This is load-bearing because the same replacement is used in the proof of Theorem 1.1 and in Lemma 3.4.
[Theorem 1.1, §3.1] The claimed asymptotic is false as stated, not merely unsupported. For odd p=2r+1, the normal form of Lemma 2.8 gives the exact generating function Σ_{x≥0}|Q_x|t^x = (1/2) G(t)/(1-G(t)) with G(t)=2Σ_{j=1}^r t^{2j+2}, up to a constant factor and finite polynomial corrections. Hence |Q_x| grows like ρ^{-x}, where ρ is the positive root of G(ρ)=1. For r=10, ρ^{-1}≈1.414, i.e. the exact exponential rate is ≈0.347. The Gaussian sum in Theorem 1.1 contains, already at n=⌊x/4⌋, a term with exponential rate (1/4)log(2r) - (1-(r+3)/4)^2/(2·(1/4)σ^2); for r=10 this equals (1/4)log20 - (9/4)^2/(16.5) ≈ 0.442, which is strictly larger than the exact rate. The theorem therefore cannot be asymptotically equivalent to |Q_x| for r=10.
[Lemma 3.4 and Theorem 1.2] Lemma 3.4 uses the same invalid CLT replacement as Lemma 3.2, so the claimed asymptotic for |Sym_x| is unsupported. In addition, the comparison |T_x|≃|Sym_x| is justified only by the short heuristic paragraph after Lemma 3.4; no proof is given that the p-reciprocal and symmetric p-reciprocal classes are exponentially negligible relative to the symmetric ones. Thus Theorem 1.2 is conditional on an unproved comparison and on a false probabilistic estimate.

minor comments (4)

[Lemma 3.2, proof] The definition of ε in the line 'for ϵ = 1/x Φ(...)' is garbled: the intended dependence on x and the choice of K should be stated explicitly, and the error term should be made uniform in a way that is actually available from the CLT.
[General] There are numerous typos and infelicities: 'recipocal', 'SuzzI Valli' for 'Suzzi Valli', 'Cauchy-Schwartz' for 'Cauchy-Schwarz', 'obtian' for 'obtain', and 'choosen from ... two ...' in the proof of Theorem 1.1.
[References] Reference [BMV] is incomplete: it lists authors but no title, venue, or year. Reference [Ma] is a thesis and should be identified as such in the bibliography.
[Theorem 3.1] Theorem 3.1 is a standard statement of the CLT and is mislabeled as a theorem; the notation 'd−→' is missing the arrow and should be written as '→ d' or '⇒'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main asymptotic is a self-contained CLT counting argument, and the cited prior results are independent theorems rather than inputs that make the prediction forced by construction.

full rationale

The paper's derivation chain is not circular. Theorem 1.1 counts reciprocal classes by combining a normal form for reciprocal words (cited from the authors' earlier [DG2, Lemma 3.2]) with a representative-per-class factor (cited from [DG1, Theorem 2.7]), and then estimates the number of admissible n-tuples by an i.i.d. CLT computation. The cited prior results are parameter-free theorems with stated assumptions that do not include the target asymptotic; they are externally falsifiable and are not fitted to the quantity being predicted. The CLT step itself is self-contained and does not assume the conclusion. The modular-group special case reproduces the independent Basmajian-Suzzi Valli result, providing an external benchmark. The serious mathematical gap identified in the manuscript—use of the CLT in a large-deviation range where the relative error of the normal approximation is uncontrolled—is a correctness issue, not a reduction of the prediction to its inputs by construction. No specific equation in the paper makes the claimed asymptotic equal to an input by definition.

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

The paper introduces no free parameters in the usual sense: the means and variances in the CLT formulas are computed from the assumed uniform distribution rather than fitted. The burden is carried by the uniform-distribution modeling assumption and by earlier self-cited structural results.

assumptions (5)

standard math Central Limit Theorem for i.i.d. random variables (Theorem 3.1)
The paper uses the CLT to approximate the distribution of sums of absolute exponent values; this is standard, but the uniformity needed in the tails is not supplied by the theorem as stated.
domain assumption Uniform distribution of exponent choices k_i in cyclically reduced reciprocal words
In Lemmas 3.2, 3.3 and the proofs of the theorems, each allowed k_i or |k_i| is assumed equally likely, so that counting tuples reduces to probability. This is a modeling choice, not a proven symmetry of reciprocal classes.
domain assumption Normal form and per-class element counts from [DG1] and [DG2] (Theorem 2.7, Lemma 2.8)
The paper assumes the classification of reciprocal word types and the exact number of symmetric and p-reciprocal elements per class from the authors' earlier papers, without proof.
ad hoc to paper Uniform CLT approximation error below the smallest tail probability
Lemma 3.2 requires the normal approximation error for P(S_n <= x) to be smaller than epsilon/2 with epsilon = (1/x)Phi((x-xmu)/(sqrt(x)sigma)); this is not a consequence of any stated theorem and is not true in the tails.
domain assumption The comparison |T_x| approximately equal to |Sym_x| for even p using [DG2] counts
Theorem 1.2 asserts that total reciprocal classes grow like symmetric classes because the p-reciprocal and symmetric-p-reciprocal counts have smaller maximum exponent; the supporting counts are cited from [DG2] and not shown.

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Cite this review

Pith. "Pith review of Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups." pith.science (2026). https://pith.science/paper/HKMC6E35

@misc{pith2026250608449,
  author       = {Pith},
  title        = {Pith review of: Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HKMC6E35}},
  note         = {Machine review of arXiv:2506.08449}
}

read the original abstract

We estimate the asymptotic growth of reciprocal conjugacy classes in Hecke groups using their free product structure and word lengths of reciprocal elements. Our approach is different from other works in this direction and uses tools from basic probability theory.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Baake, J

M. Baake, J. A. G. Roberts, The structure of reversing symmetry groups. Bull. Aust. Math. Soc. 73, No. 3 (2006), 445-459

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[2]

F. Boca, V. Pasol, A. Popa and A. Zaharescu. Pair correlation of angles between reciprocal geodesics on the modular surface. Alg. Numb. Theo. 8 (2014) 999–1035

work page 2014
[3]

Basmajian and R

A. Basmajian and R. Suzzi Valli, Combinatorial Growth in the Modular Group, Groups Geom. Dyn. 16 (2022), 683–703

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Basmajian and R

A. Basmajian and R. Suzzi Valli, Counting cusp excursions of reciprocal geodesics, In: Geometry, groups and mathematical philosophy, 21–30. Contemp. Math., 811 American Mathematical Society, [Providence], RI, 2025

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Basmajian, R

A. Basmajian, R. Suzzi Valli, B. T. Marmolejo,

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Bourgain and A

J. Bourgain and A. Kontorovich, Beyond expansion, III: Reciprocal geodesics. Duke Math. J.168 (2019), no.18, 3413–3435

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D. Das and K. Gongopadhyay. Reciprocity in the Hecke Groups. New York J. Math. 30 (2024) 722–744

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[9]

Das and K

D. Das and K. Gongopadhyay. Spherical growth of reciprocal classes in the Hecke Groups. arXiv:2411.00739

arXiv
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Erlandsson and J

V. Erlandsson and J. Souto, Counting and equidistribution of reciprocal geodesics and dihedral groups. Int. Math. Res. Not. (2024), no. 13, 10298–10318

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B. T. Marmolejo, Growth of Conjugacy Classes of Reciprocal Words in Triangle Groups(2020). CUNY Academic Works. https://academicworks.cuny.edu/gc_etds/3972

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A. G. O’Farrell and I. Short, Reversibility in Dynamics and Group Theory . London Mathematical Society Lecture Note Series, vol.416, Cambridge University Press, Cambridge, 2015

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Parkkonen and F

J. Parkkonen and F. Paulin, Counting and equidistribution of strongly reversible closed geodesics in negative curvature . arXiv:2505.07738

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Baake, J

M. Baake, J. A. G. Roberts, The structure of reversing symmetry groups. Bull. Aust. Math. Soc. 73, No. 3 (2006), 445-459

work page 2006

[2] [2]

F. Boca, V. Pasol, A. Popa and A. Zaharescu. Pair correlation of angles between reciprocal geodesics on the modular surface. Alg. Numb. Theo. 8 (2014) 999–1035

work page 2014

[3] [3]

Basmajian and R

A. Basmajian and R. Suzzi Valli, Combinatorial Growth in the Modular Group, Groups Geom. Dyn. 16 (2022), 683–703

work page 2022

[4] [4]

Basmajian and R

A. Basmajian and R. Suzzi Valli, Counting cusp excursions of reciprocal geodesics, In: Geometry, groups and mathematical philosophy, 21–30. Contemp. Math., 811 American Mathematical Society, [Providence], RI, 2025

work page 2025

[5] [5]

Basmajian and Mingkun Liu, Low lying geodesics on the modular surface and necklaces, Mathematical research letters, to appear

A. Basmajian and Mingkun Liu, Low lying geodesics on the modular surface and necklaces, Mathematical research letters, to appear

work page

[6] [6]

Basmajian, R

A. Basmajian, R. Suzzi Valli, B. T. Marmolejo,

work page

[7] [7]

Bourgain and A

J. Bourgain and A. Kontorovich, Beyond expansion, III: Reciprocal geodesics. Duke Math. J.168 (2019), no.18, 3413–3435

work page 2019

[8] [8]

Das and K

D. Das and K. Gongopadhyay. Reciprocity in the Hecke Groups. New York J. Math. 30 (2024) 722–744

work page 2024