{"id":"3ff690a7-0614-46e5-bbd7-ca5d4b980b96","arxiv_id":"1908.05321","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each k at least 2, the conjugacy growth series of BS(1,k) with respect to the standard generating set is transcendental, and its growth rate equals the standard growth rate.","lead":"This paper computes the number of conjugacy classes of each word length in the soluble Baumslag-Solitar groups BS(1,k), and shows the resulting generating series are transcendental. It also proves that conjugacy growth and ordinary word growth have the same exponential rate, a property previously seen mainly in hyperbolic and acylindrically hyperbolic groups.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader identified the asymptotic counting in Proposition 22 and the applicability of Flajolet's Theorem D as the main point to verify. I agree that this is the step on which the transcendence conclusion rests. However, stress-testing shows the step is secure: the two-sided bounds (11) and (12) follow directly from the syllable-count bounds without needing a quantitative estimate for periodic words, and the coefficient growth Θ(ρ^{-n}/n) cannot occur for an algebraic series. The derivative argument provides a robust route even if the cited theorem's precise hypotheses are in question. The exact cycle formulas in Section 6 are internally consistent, including the subtraction of the excluded alternating families. I therefore see no change to the reader's ACCEPT verdict, though a direct check of the cited theorem or a numerical verification of the coefficient bounds would still be a worthwhile safeguard.","tokens_in":21220,"tokens_out":36195,"duration_ms":359817,"concrete_test":"For a small case such as k=3, compute the first 200 coefficients of -Σ_{j≥1} φ(j)/j log(1 - S_o(z^j)) from Section 6 and verify they lie between C1 ρ^{-n}/n and C2 ρ^{-n}/n with ρ_o the dominant root of 1 - S_o(z). Then independently check that zF'(z) would have coefficients Θ(ρ^{-n}) if F were algebraic, which forces a pole and hence a logarithmic singularity for F, confirming non-algebraicity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central argument. The weakest point is Proposition 22's use of Flajolet's Theorem D: the paper proves two-sided bounds cρ^{-n}/n ≤ a_n ≤ Cρ^{-n}/n rather than an exact asymptotic, and it does not quantify the periodic-word correction. These bounds are nonetheless sufficient. For every word with m syllables, the cyclic-orbit size is at least n/(r+1) and at most n, so the number of cyclic representatives is automatically sandwiched between W_n/n and (r+1)W_n/n; periodicity is irrelevant to the inequalities. The resulting Θ(ρ^{-n}/n) coefficient growth is incompatible with algebraicity: if F were algebraic, then zF'(z) would be algebraic with coefficients Θ(ρ^{-n}), forcing a simple pole at ρ and hence a logarithmic singularity for F, a contradiction. The cycle-construction formulas in Section 6 also account correctly for periodic words via the Euler totient terms, and the subtraction of the excluded families is consistent with the proofs of Lemmas 19 and 20.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21324,"tokens_out":12244,"duration_ms":128612,"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":[{"comment":"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.","section":"Section 6, Eqs. (13)-(14) and final paragraph"},{"comment":"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.","section":"Proposition 22, inequalities (11)-(12)"}],"minor_comments":[{"comment":"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":"Proposition 22, definition of S_e"},{"comment":"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.","section":"Section 6, excluded-set notation"},{"comment":"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.","section":"Corollary 21(2) and Proposition 22"}],"recommendation":"major_revision","confidential_remarks":"The missing factor of two in Section 6 is a concrete, checkable error in the advertised formulas, not merely a presentational issue. The central transcendence and rate-equality theorems appear sound, and the normal-form analysis is careful. I would be happy to accept once the factor is corrected and the short periodic-word estimate in Proposition 22 is added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the paper that finally computes the conjugacy growth series for BS(1,k), and the main theorem is right. It deserves a serious referee; the only part I would want tightened is the step in Proposition 22 that passes from word counts to conjugacy-class counts.\n\nWhat is actually new: Breuillard–Cornulier had exponential growth but no series; Collins–Edjvet–Gill had the standard growth series. Here the authors give geodesic conjugacy representatives, compute the part coming from the base subgroup exactly as a rational series using unambiguous context-free grammars, and express the non-base part with the cycle construction. The transcendence of the full series (Corollary 23) and the equality of conjugacy and standard growth rates (Corollary 24) are both new and are natural benchmark results. The citation pattern is fine, and there is no circularity: the analytic-combinatorics input is external and the standard growth rate is computed independently.\n\nWhere the soft spots are: Proposition 22 is compressed. The text goes from 'periodic words are negligible' to the two-sided bounds Cρ^{-n}/n ≤ a_n ≤ C'ρ^{-n}/n without quantifying the error. The conclusion is correct: the lower bound is a general orbit-size argument, and the upper bound is the usual primitive-necklace estimate. But as written it is a gap in presentation, not in the underlying mathematics. One detail in the stress-test note is wrong—it says every cyclic orbit has size at least n/(r+1), which fails for t^m—so I would not rely on that version of the defense. Also, the use of Flajolet's Theorem D is terse; a referee should check that the coefficient bounds satisfy its hypotheses, since the paper cites it as a black box.\n\nWho is this for: people working on growth series, conjugacy growth, or soluble groups. It is a solid computational paper, not a conceptual revolution. I would cite it in the area and would take it to reading group. I recommend sending it to peer review, with a request to expand the proof of Proposition 22.","headline":"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.","tokens_in":21895,"tokens_out":6641,"would_cite":true,"duration_ms":72639,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20E45","05E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["conjugacy growth","Baumslag-Solitar groups","transcendental generating functions","geodesic conjugacy representatives","context-free grammars","growth rates","soluble groups","asymptotic enumeration"],"falsifier":"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.","tokens_in":20974,"feed_emoji":"🧮","tokens_out":6503,"duration_ms":63257,"temperature":0.7,"pith_summary":"The paper establishes sharp asymptotics for conjugacy growth in the soluble Baumslag-Solitar groups BS(1,k), k≥2, with respect to the standard generating set {a,t}. Its main results are that the conjugacy growth series is transcendental and that the conjugacy growth rate equals the standard growth rate. The proof works by giving a complete list of geodesic conjugacy representatives: conjugacy classes inside the base normal subgroup Z_k have rational growth series computed from an unambiguous context-free grammar, while classes with nonzero t-exponent are counted by cyclic equivalence classes of syllable words, whose asymptotics are of the form constant times $ρ^{{-n}}$/n. This sharpens earlier exponential-growth results for soluble groups and supports the conjecture that non-virtually-abelian finitely presented groups have transcendental conjugacy growth series.","feed_headline":"Conjugacy growth of Baumslag-Solitar groups is transcendental","feed_subtitle":"New geodesic representatives give a full count of conjugacy classes and equal conjugacy and word growth rates.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the geodesic normal forms for elements of BS(1,k) and the standard growth-series polynomials whose denominators later dominate both growth rates.","marker":"[8]"},{"why":"Provides the theorem that unambiguous context-free languages have algebraic growth series, underpinning the rational series for the base subgroup.","marker":"[4]"},{"why":"Supplies the transcendence criterion (Theorem D) for generating functions with coefficients bounded by multiples of ρ^{-n}/n.","marker":"[12]"},{"why":"Provides the cycle construction and analytic combinatorics formalism used to write the conjugacy series outside Z_k as the logarithmic sums in (13)-(14).","marker":"[13]"},{"why":"Establishes exponential conjugacy growth for soluble non-virtually-nilpotent groups, the qualitative result this paper sharpens to asymptotics.","marker":"[2]"},{"why":"Gives the formal conjugacy growth setup and the asymptotic form constant·α^n/n in acylindrically hyperbolic groups, used as comparison and motivation.","marker":"[1]"},{"why":"Puts forward the conjecture that rational conjugacy growth series characterise virtually abelian groups, which the transcendental result supports.","marker":"[11]"},{"why":"Supplies the standard growth rate computation for BS(1,k) used in Corollary 24 to identify the growth-rate polynomial.","marker":"[3]"}],"fun_headline_variants":["Baumslag-Solitar conjugacy growth: transcendental series, equal rates","Conjugacy growth of BS(1,k) is transcendental, rates match","Geodesic reps reveal transcendental conjugacy growth in BS(1,k)","Conjugacy growth series transcendental for soluble Baumslag-Solitar","Equal conjugacy and word growth rates in BS(1,k) groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Baumslag-Solitar conjugacy growth: transcendental series, equal rates","Conjugacy growth of BS(1,k) is transcendental, rates match","Geodesic reps reveal transcendental conjugacy growth in BS(1,k)","Conjugacy growth series transcendental for soluble Baumslag-Solitar","Equal conjugacy and word growth rates in BS(1,k) groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2634,"prompt_tokens":828,"completion_tokens":1806,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":444,"completion_tokens_details":{"reasoning_tokens":1708}},"tokens_in":444,"tokens_out":1806,"duration_ms":12426,"temperature":1.0,"reasoning_tokens":1708,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:17:36.329312+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Collins, M","cited_arxiv_id":null,"evidence_quote":"Supplies the geodesic normal forms for elements of BS(1,k) and the standard growth-series polynomials whose denominators later dominate both growth rates."},{"cited_title":"Chomsky, and M","cited_arxiv_id":null,"evidence_quote":"Provides the theorem that unambiguous context-free languages have algebraic growth series, underpinning the rational series for the base subgroup."},{"cited_title":"Flajolet, Analytic models and ambiguity of context-free languages , Theoretical Computer Science, 49 (1987), 283–309","cited_arxiv_id":null,"evidence_quote":"Supplies the transcendence criterion (Theorem D) for generating functions with coefficients bounded by multiples of ρ^{-n}/n."},{"cited_title":"Flajolet and R","cited_arxiv_id":null,"evidence_quote":"Provides the cycle construction and analytic combinatorics formalism used to write the conjugacy series outside Z_k as the logarithmic sums in (13)-(14)."},{"cited_title":"Breuillard, and Y","cited_arxiv_id":null,"evidence_quote":"Establishes exponential conjugacy growth for soluble non-virtually-nilpotent groups, the qualitative result this paper sharpens to asymptotics."},{"cited_title":"Formal conjugacy growt h in acylindrically hyperbolic groups","cited_arxiv_id":null,"evidence_quote":"Gives the formal conjugacy growth setup and the asymptotic form constant·α^n/n in acylindrically hyperbolic groups, used as comparison and motivation."},{"cited_title":"Rational Growth in Virtually Abelian Groups","cited_arxiv_id":"1808.06371","evidence_quote":"Puts forward the conjecture that rational conjugacy growth series characterise virtually abelian groups, which the transcendental result supports."},{"cited_title":"Bucher, and A","cited_arxiv_id":null,"evidence_quote":"Supplies the standard growth rate computation for BS(1,k) used in Corollary 24 to identify the growth-rate polynomial."}],"review_version":1}