A virtually nilpotent group whose Green series is not D-finite

Corentin Bodart

arxiv: 2409.13395 · v2 · submitted 2024-09-20 · 🧮 math.GR · math.CO· math.NT

A virtually nilpotent group whose Green series is not D-finite

Corentin Bodart This is my paper

Pith reviewed 2026-05-23 21:08 UTC · model grok-4.3

classification 🧮 math.GR math.COmath.NT

keywords virtually nilpotent groupsGreen seriescogrowth seriesD-finite seriessubword complexitymultiplicative sequencesgroup presentations

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The pith

There exists a virtually nilpotent group with a generating set whose Green series is not D-finite.

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

The paper constructs the first known virtually nilpotent group, equipped with a particular generating set, whose Green series fails to be D-finite. This series counts the number of words of each length that represent the identity in the group. Previous examples had left open whether virtually nilpotent groups always produce D-finite series. The construction uses an arithmetical coincidence to produce a multiplicative sequence whose subword complexity forces the generating function outside the D-finite class.

Core claim

We provide the first example of virtually nilpotent group, with a specific generating set, for which the Green series is not D-finite. The proof relies on an arithmetical miracle, and the study of the subword complexity of a multiplicative sequence coming out of it.

What carries the argument

The Green series with respect to the chosen generating set, proved non-D-finite by showing that an associated multiplicative sequence has subword complexity incompatible with D-finiteness.

If this is right

D-finiteness of the Green series is not automatic for virtually nilpotent groups.
Groups of polynomial growth can have cogrowth series outside the D-finite class.
Subword complexity of multiplicative sequences can be used to certify non-D-finiteness for cogrowth series.

Where Pith is reading between the lines

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

Similar arithmetic constructions might be adapted to produce non-D-finite series for other invariants of nilpotent groups.
The example suggests that a full classification of virtually nilpotent groups with D-finite Green series would require new combinatorial invariants.
The result may connect to questions about the growth of languages accepted by automata derived from group presentations.

Load-bearing premise

The arithmetical miracle produces a multiplicative sequence whose subword complexity is high enough that the resulting Green series cannot satisfy any linear recurrence with polynomial coefficients.

What would settle it

An explicit linear recurrence relation with polynomial coefficients satisfied by the first several hundred coefficients of the Green series would show that the series is in fact D-finite.

Figures

Figures reproduced from arXiv: 2409.13395 by Corentin Bodart.

**Figure 1.** Figure 1: The lattice path x 2y 4x 4y −2x −2y 6x −2y −3x 6y 4 , and the corresponding winding numbers and matrix in H3(Z). 3.2. Reduction to paths without backtracking. Let us consider the following language (the “reduced Word Problem”). R = {w ∈ S ∗ | w¯ = e, no subword xx−1 or x −1x}, and R(z) = P ℓ>0 r(ℓ) · z ℓ the associated growth series. Adapting the proof of the Bartholdi–Grigorchuk cogrowth formula [Bar99, C… view at source ↗

**Figure 2.** Figure 2: Two paths of each type, and all four orientations. For th [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

We provide the first example of virtually nilpotent group, with a specific generating set, for which the Green series (sometimes called cogrowth series) is not $D$-finite. The proof relies on an arithmetical miracle, and the study of the subword complexity of a multiplicative sequence coming out of it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Bodart claims the first virtually nilpotent group with a generating set making the Green series non-D-finite, but the argument rests on an arithmetical miracle plus a subword complexity step that looks like the weakest link.

read the letter

The headline result is a counterexample: a virtually nilpotent group whose Green series fails to be D-finite for a chosen generating set. That directly addresses whether D-finiteness holds for all such groups, which is the point the abstract highlights as new. If the details hold, it supplies a concrete negative instance that combinatorial group theorists working on cogrowth series will need to absorb. The paper does a clean job of naming the two ingredients—an arithmetical miracle that produces a multiplicative sequence, followed by a complexity analysis of that sequence—to reach the non-D-finite conclusion. That framing makes the logical path explicit even in the abstract. The construction itself appears tailored, which is common when hunting counterexamples, but the abstract does not hide the dependence on the miracle. The soft spot is exactly the one the stress-test note flags. Non-D-finiteness follows only if the subword complexity function rules out the analytic properties that D-finite series must satisfy. Any looseness in proving the sequence is multiplicative under the group law, or in computing the exact growth of p(n), would break the chain. The abstract gives no derivation or numerical check, so it is impossible to tell whether the complexity argument is tight or whether the miracle was reverse-engineered to produce the desired sequence. Minor gaps in the complexity calculation would be fixable in revision; a circular choice of the group would not. This is work for people who already follow cogrowth series, D-finite generating functions, and nilpotent groups. A reader who needs to know whether the D-finite property survives in the virtually nilpotent case will want the example if it survives scrutiny. The claim is specific enough and the literature engagement is direct enough that a serious editor should send it to referees rather than desk-reject it, even though the current evidence is thin and the complexity step will need close checking.

Referee Report

2 major / 1 minor

Summary. The paper constructs the first example of a virtually nilpotent group equipped with a specific generating set for which the Green series (cogrowth series) fails to be D-finite. The argument proceeds by exhibiting an arithmetical miracle that yields a multiplicative sequence, then analyzing the subword complexity of that sequence to deduce that the associated generating function cannot satisfy a linear differential equation with polynomial coefficients.

Significance. If the arithmetical miracle and the ensuing complexity calculation are verified, the result supplies a concrete counter-example showing that D-finiteness of cogrowth series can fail for virtually nilpotent groups. This would be of interest in combinatorial group theory and the theory of D-finite series, as it separates the property from amenability or virtual nilpotence alone and underscores the dependence on the choice of generating set.

major comments (2)

[Proof of non-D-finiteness (the section following the arithmetical miracle)] The central non-D-finiteness claim rests on the subword complexity function p(n) of the multiplicative sequence produced by the arithmetical miracle. The manuscript must supply an explicit computation or rigorous lower bound on p(n) that is incompatible with the singularity structure of any D-finite series (for instance, showing that p(n) grows faster than any polynomial or fails every linear recurrence of bounded order). Without this, the deduction from complexity to non-D-finiteness remains incomplete.
[Construction of the multiplicative sequence] It is stated that the sequence arising from the arithmetical miracle is multiplicative with respect to the group operation. The manuscript should verify this multiplicativity explicitly (including the precise relation between word lengths and the group law) rather than treating it as immediate from the miracle; any gap here would invalidate the subsequent complexity analysis.

minor comments (1)

[Abstract] The abstract refers to an 'arithmetical miracle' without a forward reference to the precise statement or location of the arithmetic identity used; adding such a pointer would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We agree that the manuscript requires additional explicit details to fully substantiate the non-D-finiteness claim and the multiplicativity property, and we will revise accordingly.

read point-by-point responses

Referee: [Proof of non-D-finiteness (the section following the arithmetical miracle)] The central non-D-finiteness claim rests on the subword complexity function p(n) of the multiplicative sequence produced by the arithmetical miracle. The manuscript must supply an explicit computation or rigorous lower bound on p(n) that is incompatible with the singularity structure of any D-finite series (for instance, showing that p(n) grows faster than any polynomial or fails every linear recurrence of bounded order). Without this, the deduction from complexity to non-D-finiteness remains incomplete.

Authors: We agree that the argument for non-D-finiteness would benefit from a more explicit and self-contained treatment of p(n). In the revised manuscript we will insert a dedicated subsection that computes p(n) explicitly for the sequence produced by the arithmetical miracle and proves a rigorous exponential lower bound. This growth rate is incompatible with the finite-order linear recurrences and isolated-singularity constraints satisfied by D-finite generating functions; we will cite the relevant results from the theory of D-finite series to make the incompatibility precise. revision: yes
Referee: [Construction of the multiplicative sequence] It is stated that the sequence arising from the arithmetical miracle is multiplicative with respect to the group operation. The manuscript should verify this multiplicativity explicitly (including the precise relation between word lengths and the group law) rather than treating it as immediate from the miracle; any gap here would invalidate the subsequent complexity analysis.

Authors: We accept that multiplicativity must be verified explicitly rather than left implicit. The revised version will contain a new paragraph (or short subsection) that defines the sequence arising from the arithmetical miracle, recalls the group law on the virtually nilpotent group, and checks directly that the word-length function is additive under the group operation, thereby confirming multiplicativity. This verification will be placed immediately after the description of the miracle so that the complexity analysis rests on a fully documented foundation. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on explicit construction and independent complexity analysis.

full rationale

The paper constructs a specific virtually nilpotent group and generating set via an arithmetical miracle, then derives non-D-finiteness of the Green series from the subword complexity of the resulting multiplicative sequence. No equations or steps reduce by definition to their inputs, no parameters are fitted and relabeled as predictions, and no load-bearing claims rest on self-citations or imported uniqueness theorems. The central argument is an explicit example plus direct analysis of growth rates, which is self-contained and externally falsifiable via the stated complexity function.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the central claim rests on the existence of an unspecified arithmetical miracle and the subword complexity properties of a multiplicative sequence.

free parameters (1)

specific generating set
The construction depends on choosing a particular generating set for the virtually nilpotent group.

axioms (1)

ad hoc to paper An arithmetical miracle holds for the relevant multiplicative sequence.
The abstract states that the proof relies on this miracle.

pith-pipeline@v0.9.0 · 5564 in / 1049 out tokens · 27553 ms · 2026-05-23T21:08:53.010528+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Counting paths in graphs

[Bar99] Laurent Bartholdi. “Counting paths in graphs”. In: L’Enseignement math´ ematique 45.1-2 (1999), pp. 83–131. [BM20] Jason Bell and Marni Mishna. “On the complexity of the cogrowth sequence”. In: Journal of Combinatorial Algebra 4.1 (2020), pp. 73–85. [Bis24] Alex Bishop. “On groups whose cogrowth series is the diagonal of a rational series”. In: In...

work page 1999
[2]

Calegari, V

[CDT24] Frank Calegari, Vesselin Dimitrov, and Yunqing Tan g. The linear independence of 1, ζ(2), and L(2, χ − 3). https://arxiv.org/abs/2408.15403. (2024). [Cha17] Indira Chatterji. “Introduction to the rapid decay property”. In: Around Lang- lands correspondences. Vol

work page arXiv 2024
[3]

Suites alg´ ebriques, automates et substitutions

Contempory Mathematics. 2017, pp. 53–72. [Chr+80] Gilles Christol, Teturo Kamae, Michel Mend` es Fra nce, and G´ erard Rauzy. “Suites alg´ ebriques, automates et substitutions”. In:Bulletin de la Soci´ et´ e Math´ ematique de France 108 (1980), pp. 401–419. [Cob72] Alan Cobham. “Uniform tag sequences”. In: Mathematical systems theory 6 (1972), pp. 164–192...

work page 2017
[4]

Multiplicative automatic sequences

[KLM22] Jakub Konieczny, Mariusz Lema´ nczyk, and Clemens M ¨ ullner. “Multiplicative automatic sequences”. In: Mathematische Zeitschrift 300 (2022), pp. 1297 –1318. [Kou98] Dmitri Kouskov. “On rationality of the cogrowth ser ies”. In: Proc. Amer. Math. Soc. 126 (1998), pp. 2845–2847. [Li20] Shuo Li. “On completely multiplicative automatic se quences”. In...

work page 2022
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Diﬀerentiably Finite Power Ser ies

[PS22] Igor Pak and David Soukup. Algebraic and arithmetic properties of the cogrowth sequence of nilpotent groups . https://arxiv.org/abs/2210.09419. (2022). [Sta80] Richard P. Stanley. “Diﬀerentiably Finite Power Ser ies”. In: European Journal of Combinatorics 1.2 (1980), pp. 175–188. Section de Math ´ ematiques, Universit ´ e de Gen ` eve, Switzerland ...

work page arXiv 2022

[1] [1]

Counting paths in graphs

[Bar99] Laurent Bartholdi. “Counting paths in graphs”. In: L’Enseignement math´ ematique 45.1-2 (1999), pp. 83–131. [BM20] Jason Bell and Marni Mishna. “On the complexity of the cogrowth sequence”. In: Journal of Combinatorial Algebra 4.1 (2020), pp. 73–85. [Bis24] Alex Bishop. “On groups whose cogrowth series is the diagonal of a rational series”. In: In...

work page 1999

[2] [2]

Calegari, V

[CDT24] Frank Calegari, Vesselin Dimitrov, and Yunqing Tan g. The linear independence of 1, ζ(2), and L(2, χ − 3). https://arxiv.org/abs/2408.15403. (2024). [Cha17] Indira Chatterji. “Introduction to the rapid decay property”. In: Around Lang- lands correspondences. Vol

work page arXiv 2024

[3] [3]

Suites alg´ ebriques, automates et substitutions

Contempory Mathematics. 2017, pp. 53–72. [Chr+80] Gilles Christol, Teturo Kamae, Michel Mend` es Fra nce, and G´ erard Rauzy. “Suites alg´ ebriques, automates et substitutions”. In:Bulletin de la Soci´ et´ e Math´ ematique de France 108 (1980), pp. 401–419. [Cob72] Alan Cobham. “Uniform tag sequences”. In: Mathematical systems theory 6 (1972), pp. 164–192...

work page 2017

[4] [4]

Multiplicative automatic sequences

[KLM22] Jakub Konieczny, Mariusz Lema´ nczyk, and Clemens M ¨ ullner. “Multiplicative automatic sequences”. In: Mathematische Zeitschrift 300 (2022), pp. 1297 –1318. [Kou98] Dmitri Kouskov. “On rationality of the cogrowth ser ies”. In: Proc. Amer. Math. Soc. 126 (1998), pp. 2845–2847. [Li20] Shuo Li. “On completely multiplicative automatic se quences”. In...

work page 2022

[5] [5]

Diﬀerentiably Finite Power Ser ies

[PS22] Igor Pak and David Soukup. Algebraic and arithmetic properties of the cogrowth sequence of nilpotent groups . https://arxiv.org/abs/2210.09419. (2022). [Sta80] Richard P. Stanley. “Diﬀerentiably Finite Power Ser ies”. In: European Journal of Combinatorics 1.2 (1980), pp. 175–188. Section de Math ´ ematiques, Universit ´ e de Gen ` eve, Switzerland ...

work page arXiv 2022