REVIEW 2 major objections 3 minor 19 references
The conjugacy growth of the soluble Baumslag-Solitar groups
T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The conjugacy growth series of the soluble Baumslag-Solitar groups BS(1,k) is transcendental, and the conjugacy growth rate coincides with the standard growth rate.
desk verdict A careful, correct computation of conjugacy growth for BS(1,k); the main theorem stands, and the paper deserves a serious referee, though Proposition 22's cyclic-counting step is compressed. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the set A of conjugacy geodesics modulo cyclic permutation, together with its syllable decomposition. For a class with t-exponent m>0, every conjugacy geodesic has, up to cyclic permutation, the form $a^{{x_0}}$ t $a^{{x_1}}$ t ⋯ $a^{{x_{m-1}}$} t, with the exponents restricted by |x_i|≤r in the odd case or by finer constraints that forbid certain transitions around ±r in the even case, and with specified periodic words excluded. Classes in Z_k have representatives given by explicit word forms Oa–Od, Ea–Ed, or 2a–2d. The machinery is twofold: an unambiguous context-free grammar produces the Z_k representatives, so the classical theorem on unambiguous context-free languages gives algebraicity, and the grammar-to-equations method actually yields a rational series; for the outside classes, the syllable set S_o or S_e has a rational generating function, and the cycle construction expresses the series up to cyclic permutation as a logarithmic sum, after which [12, Theorem D] converts the coefficient bounds into transcendence.
What would settle it
Compute the exact numbers a_n of cyclic representatives of length n from the cycle generating functions (13)-(14) for a fixed k, say k=2 or k=3, and compare a_n to $ρ^{{-n}}$/n: if the ratio tends to 0 or ∞ along a subsequence, the asymptotic bounds claimed in Proposition 22 are false. A direct enumeration of cyclic representatives for n up to a few hundred would suffice to check whether the ratio remains between the asserted constants c and C.
Extended reading notes
Core claim
For each k≥2, the paper proves that the strict conjugacy growth series C(z)=Σ c(n)z^n of BS(1,k) with respect to {a,t} splits as a rational series for the classes lying in Z_k plus a transcendental series for all other classes. The rational part is given explicitly in formulas (3), (4), and (5) for odd k, even k>2, and k=2 respectively. The transcendental part arises because the number of cyclic representatives of length n among the conjugacy geodesics with t-exponent m≠0 is asymptotically sandwiched between positive multiples of $ρ^{{-n}}$/n, for a root ρ of an explicit polynomial; by the transcendence criterion of [12, Theorem D], any such sequence has a transcendental generating function. Consequently the whole conjugacy growth series is transcendental (Corollary 23), and the smallest positive singularity of the series is the same one that determines the standard growth rate, so conjugacy and standard growth rates are equal (Corollary 24).
Load-bearing premise
The proof's transcendence step relies on the unquantified claim in Proposition 22 that periodic words and words excluded by local constraints contribute negligibly, so that for every large n the number of cyclic representatives is bounded between two positive multiples of $ρ^{{-n}}$/n; if that bound fails at some scale, the cited transcendence criterion no longer applies.
Editorial extensions
If this is right
- The full conjugacy growth series of BS(1,k) is transcendental for every k≥2 with respect to the generating set {a,t}.
- The conjugacy growth rate equals the word growth rate for {a,t}; for k=2 this rate is approximately 1/0.590, and for larger k it is the reciprocal of the dominant root of an explicit polynomial.
- The number of conjugacy classes of length n grows like a constant times ρ^{-n}/n, so conjugacy classes are asymptotically fewer than group elements by a factor of 1/n.
- The explicit series formulas (13)-(14) give a concrete, though transcendental, description of conjugacy growth outside the base subgroup, from which numerical coefficients can be extracted.
- These results provide further evidence for the conjecture that conjugacy growth series are rational only for virtually abelian groups.
Reading between the lines
- If the same geodesic-representative method can be adapted to other generating sets, the conjecture that transcendence is independent of the generating set would follow for BS(1,k).
- The exact cycle formulas (13)-(14) allow a numerical test of the asserted ρ^{-n}/n asymptotics: one could compute c(n) for moderate n and check whether the ratio stays within the asserted positive constants, which would also test the unquantified negligibility of periodic representatives.
- The equal-rates phenomenon may extend to more general metabelian one-relator groups whose standard growth denominator shares a dominant root with the conjugacy counting function; the BS(1,k) proof isolates this mechanism.
- The complete geodesic representative set A could serve as a normal form for other algorithmic questions in BS(1,k), such as geodesic language classification or random walks.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies conjugacy growth of the soluble Baumslag-Solitar groups BS(1,k), k >= 2, with respect to the standard generating set {a,t}. It gives a complete description of geodesic conjugacy representatives: for the base subgroup Z_k the representatives form an unambiguous context-free language whose growth series is computed explicitly and shown to be rational, and for the remaining conjugacy classes (those with nonzero t-exponent sum) the representatives are described as cycles in finite sets of syllables, following normal-form results of Collins-Edjvet-Gill. The paper then derives two-sided coefficient asymptotics for the positive-exponent representatives and uses Flajolet's Theorem D to conclude that the generating function for these classes is transcendental, hence, since the base-subgroup series is rational, the full conjugacy growth series is transcendental (Corollary 23). It also compares the dominant singularity with the standard growth series and proves equality of the conjugacy and standard growth rates (Corollary 24). Section 6 gives explicit formulas for the conjugacy growth series using the cycle construction from analytic combinatorics. The main results confirm conjectures about rational/transcendental conjugacy growth series and about equality of conjugacy and standard growth rates.
Significance. If the results are correct, this is a substantial contribution to the conjugacy growth theory of soluble groups. The paper provides the first complete conjugacy-representative description and exact generating-function formulas for these groups, and it establishes the expected transcendental behavior and rate equality for a non-acylindrically hyperbolic family. The proofs are detailed and grounded in established machinery: Chomsky-Schutzenberger for unambiguous context-free languages, the DSV method for explicit rational series, Collins-Edjvet-Gill normal forms for geodesics, and the Flajolet-Sedgewick cycle construction. The explicit computations of the rational base-subgroup series and the explicit descriptions of the cyclic representatives are valuable in themselves and make the main claims checkable. The central transcendence and rate-equality arguments are sound in their main lines; the issues I found are local and fixable, but one of them, the missing factor of two in the Section 6 formulas, affects the advertised formulas for the conjugacy growth series.
major comments (2)
- [Section 6, Eqs. (13)-(14) and final paragraph] The displayed formulas count only the conjugacy classes represented by A_+, not by the full set A = A_+ union A_- from Corollary 21. Corollary 21 defines A as the union of the positive- and negative-exponent representatives, with a length-preserving bijection between the two parts. These two parts represent disjoint families of conjugacy classes, because the t-exponent sum is a homomorphism to Z and is invariant under conjugation. Therefore the contribution of the m != 0 classes to the full conjugacy growth series must appear with a factor 2, and the excluded geometric series No(z) or Ne(z) must also be subtracted twice. As written, the formulas in Section 6 describe only the non-negative exponent half of the conjugacy growth series. This does not affect Corollaries 23 and 24, but it does affect the claimed formulas for the series, so the section should be corrected.
- [Proposition 22, inequalities (11)-(12)] The proof of the two-sided bounds for the number of cyclic representatives relies on the statement that 'the number of powers is negligible compared to the total number of words.' The required statement is stronger than the one written: the upper bound in (11) and (12) is of the form C rho^{-n}/n, so one needs the periodic words to contribute o(rho^{-n}/n), not merely o(rho^{-n}). The needed estimate is available: a non-trivial power in A_o or A_e has block length at most n/2, so the number of such words of length n is bounded by the total number of words of length at most n/2, which is O(rho^{-n/2}); since rho < 1/2 in the odd case and rho < 1/2 for even r > 1 (and rho < 1 for r = 1), rho^{-n/2} = o(rho^{-n}/n). I recommend adding this short estimate so that Theorem D is applied to a fully justified two-sided bound.
minor comments (3)
- [Proposition 22, definition of S_e] The notation 'a^{+- r}tt' in the definition of S_e is ambiguous. Please spell out the elements of S_e explicitly, for example as t, a^j t for 1 <= |j| <= r-1, and the two-letter concatenations a^{r} t a^{j} t for 0 <= j <= r-1 and a^{-r} t a^{-j} t for 0 <= j <= r-1.
- [Section 6, excluded-set notation] In the even case the excluded set is also denoted No(z), which clashes with the odd-case notation No(z). Rename the even-case generating function, for example to Ne(z), to avoid confusion.
- [Corollary 21(2) and Proposition 22] The condition 'if x_{i-1} = +- r then 0 <= +- x_i < r' is compact but hard to parse; a short verbal statement of the two one-sided restrictions would improve readability.
Circularity Check
No significant circularity: derivation is self-contained and uses external theorems.
full rationale
The central claims (transcendence of the conjugacy growth series and equality of the conjugacy and standard growth rates) are derived from the paper's own normal-form and orbit-counting arguments combined with external results: Chomsky-Schützenberger (Theorem 5), the DSV method, Collins-Edjvet-Gill geodesic normal forms [8], the Flajolet-Sedgewick cycle construction [13], and Flajolet's Theorem D [12]. No parameter is fitted to the target quantity; the conjugacy growth series for the base subgroup Z_k is computed explicitly as a rational function from unambiguous context-free grammars, while the non-base conjugacy classes are counted by bounding cyclic-orbit sizes between c rho^{-n}/n and C rho^{-n}/n. The equality of growth rates in Corollary 24 compares these quantities with the independently cited standard growth series from [8]; the appearance of the same denominator polynomial in both contexts is the content of the theorem, not an input. Self-citations (e.g., [1], [5], [7], [11]) are contextual prior work or conjecture statements and are not load-bearing. Potential weaknesses, such as the unquantified 'negligible' periodic-word contribution in Proposition 22 or the applicability of Flajolet's Theorem D from two-sided bounds, are matters of proof rigor and do not make the derivation circular.
Assumptions & free parameters
assumptions (5)
- domain assumption Propositions 1, 2, 3 (from [8, Section 4]) give unique geodesic representatives for the elements of Z_k in BS(1,k).
- domain assumption Lemma 16 (from [8, Lemma 2.2]): geodesics in BS(1,k) satisfy the stated subword conditions.
- standard math Chomsky-Schutzenberger theorem and the DSV method convert unambiguous context-free grammars into algebraic or rational generating series.
- standard math Flajolet's Theorem D: a sequence with coefficient asymptotics bounded between positive multiples of rho^{-n}/n has a transcendental generating function.
- standard math Cycle construction (Flajolet-Sedgewick, Theorem I.1): the generating function of necklaces on a set of words is given by the logarithmic formula.
Cite this review
Pith. "Pith review of The conjugacy growth of the soluble Baumslag-Solitar groups." pith.science (2026). https://pith.science/paper/O5H7CS5X
@misc{pith2026190805321,
author = {Pith},
title = {Pith review of: The conjugacy growth of the soluble Baumslag-Solitar groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/O5H7CS5X}},
note = {Machine review of arXiv:1908.05321}
}
abstract
In this paper we give asymptotics for the conjugacy growth of the soluble Baumslag-Solitar groups $BS(1,k)$, $k\geq 2$, with respect to the standard generating set, by providing a complete description of geodesic conjugacy representatives. We show that the conjugacy growth series for these groups are transcendental, and give formulas for the series. As a result of our computation we also establish that in each $BS(1,k)$ the conjugacy and standard growth rates are equal.
Reference graph
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