Subgroups of the Group of Formal Power Series with the Big Powers Condition

A. Brudnyi

REVIEW 2 major objections 5 minor 1 cited by

Subgroups of the Group of Formal Power Series with the Big Powers Condition

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Pith's one-line read Every finitely generated fully residually free group enters G[[r]]

desk verdict A promising embedding construction with a real gap in the trivial-amalgam free product case; the centralizer-extension result and the fully residually free embedding may survive. read the letter →

arxiv 1908.04918 v1 pith:PPAQWNZQ submitted 2019-08-14 math.GR

A. Brudnyi This is my paper · ORCID

classification math.GR MSC 20E0620F38

keywords groupofformalpowerseriesbigpowersconditionfullyresiduallyfreeproductwithamalgamationcentralizerextensionCSAseparationembeddability

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

The paper proves that groups satisfying the big powers condition—a strong guarantee that products of conjugates of nontrivial elements do not accidentally collapse—can be freely combined inside the group G[[r]] of formal power series under composition, as long as the starting group already embeds there. Specifically, every amalgamated free product over a centralizer (or over the trivial subgroup) and every extension of a centralizer of such a group remains a BP-group and can be realized inside G[[r]]. As a consequence, every finitely generated fully residually free group—a group that looks free on every finite subset—including the fundamental groups of most compact Riemann surfaces, embeds into G[[r]]. This matters because G[[r]] is the natural setting for the holonomy of local differential equations, so the result turns a question of combinatorial group theory into a concrete analytic embedding.

What carries the argument

The carrying object is the exponential map exp from the Lie algebra of formal vector fields to G[[r]], together with a transcendence trick: the coefficient field C is replaced by an isomorphic copy sigma(C) so that auxiliary transcendental elements remain algebraically independent over all coefficients in the given group. Conjugating one building-block subgroup by a power of such a transcendental series produces a copy whose intersection with the other factor is exactly the centralizer at which the amalgamation happens. The identity that makes the machinery work is that two exponentials in G[[r]] commute only when their logarithms are proportional (Lemma 2.1), which gives the CSA property; combined with the separation-condition formulation of the big powers condition, this forces any word mapping to the identity to reduce and collapse.

What would settle it

To test the construction, take H0=<exp(e1)> with rational coefficients, let c=exp(s e1+$s^{2}$ e2) with s transcendental over Q, and conjugate a second copy of H0 by $c^{{-t}}$ with t algebraically independent; then compute the low-order coefficients of a reduced word such as exp(e1)$c^{{-t}}$exp(e1)c^t. If any nontrivial reduced word equals the identity series, the claimed free-product embedding fails; conversely, the absence of such collapse in all words up to a given length is what the theorem predicts.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: for any transfinite chain of groups built from G0 by taking amalgamated free products over centralizers (or trivial amalgams) and extensions of centralizers, the resulting group is a BP-group embeddable into G[[r]] if and only if G0 is. The theorem is proved by embedding each step explicitly: one factor is conjugated by a carefully chosen transcendental formal power series, and algebraic independence over the original coefficient field forces the intersection of the two factors to be exactly the prescribed centralizer. The big powers (separation) condition of the original group then rules out any hidden collapse of words, so the generated subgroup is genuinely the intended free product or centralizer extension. The corollary the paper highlights is that every finitely generated fully residually free group is a BP-group and embeds into G[[r]], answering the question of whether surface group fundamental groups admit such embeddings.

Load-bearing premise

The load-bearing premise is that the separation condition can be applied to words involving the auxiliary transcendental series c after replacing its exponent by an integer n; this requires those conjugated words to belong to the original BP-subgroup H0, but c was chosen outside H0's coefficient field. If that containment fails, the trivial-amalgam free product case in the main theorem is not established.

Editorial extensions

If this is right

Every finitely generated fully residually free group is a BP-group and embeds into G[[r]].
The fundamental groups of all non-exceptional compact Riemann surfaces embed into G[[r]], resolving the motivating question about holonomy groups.
The Lyndon completion of a finitely generated fully residually free group also embeds into G[[r]] and is a BP-group.
When the initial group has size below the continuum, the same structural conclusion holds inside the subgroup of series with real coefficients.

Reading between the lines

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

Editorial inference: if the embedding is as explicit as the proof suggests, the same conjugation trick may embed similar amalgams and centralizer extensions in other prounipotent groups with a CSA structure, so the class of embeddable groups may be much larger than fully residually free groups.
Editorial inference: the paper leaves open whether G[[r]] itself is a BP-group; if it is, the 'if and only if' in Theorem 1.2 would become unconditional, and the entire universe of groups built from any BP-subgroup would automatically embed.
Editorial inference: the promised differential-equation application suggests testing the center property for Abel equations through the BP property of their holonomy groups—if a holonomy group fails the big powers condition, the corresponding center problem should have no formal first integral.

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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

The paper's real news is a new embedding construction: using transcendence bases of C to conjugate subgroups of G[[r]] by formal power series with transcendental coefficients, Brudnyi proves that centralizer extensions of BP-subgroups of G[[r]] are again BP and embeddable. That part (Theorem 3.1(b), Lemma 3.6) looks sound, and it yields the advertised corollary that every finitely generated fully residually free group embeds into G[[r]], answering Problem 4.15 of [Br]. The argument is genuinely new.

The soft spot is the trivial-amalgam free product case (C={1}, Lemma 3.5). After replacing the transcendental parameter t by an integer n, the proof applies the separation condition of H0 to a word like h1 c^{-n} h2 c^n ... h_{2k} c^n = 1. But c = exp(s e1 + s^2 e2) was chosen outside G_{σ(C)}[[r]], and H0 ≤ G_{σ(C)}[[r]], so c^n is not in H0 for n≠0. The word is not known to lie in H0, so the separation condition cannot be invoked. This is a real gap, not cosmetic. Lemma 3.4 has a related sticky spot where the separation condition is mentioned in passing, though the algebraic-independence argument may cover that case. In any case, Theorem 3.1(a) for C={1}, and hence Theorem 1.2 in full generality and Example 1.3(1), are unproved as written.

The centralizer-extension part is unaffected, so the fully residually free embedding may still stand. The paper's headline theorem, however, overclaims as stated.

The citation pattern is normal: [KMS] for BP facts, [MR]/[KM] for limit groups, [Br] for the problem. No circularity.

Who is this for? People working on BP groups, formal diffeomorphisms, and the center problem. A serious referee should see this because the construction is promising and likely repairable — one might choose a different c inside a larger BP group containing H0, or find another way to certify the separation step. I'd send it to review with a request to fix the trivial-amalgam case. If it can't be fixed, the paper still has a valuable sound core.

Referee Report

2 major / 5 minor

Summary. The paper studies discrete subgroups of the formal power series group G[[r]] under composition, focusing on subgroups satisfying the big powers (BP) condition. The main result, Theorem 1.2, asserts that for a chain of groups built from free products with amalgamation over centralizers and from centralizer extensions, the terminal group G_delta is a BP-group embeddable into G[[r]] if and only if the initial group G0 is. The proof proceeds by a transfinite construction, with the base case given in Theorem 3.1: (a) an amalgamated free product H1 *C H2 (with C either trivial or a common centralizer) of BP-subgroups of G[[r]] is BP and embeddable, and (b) a centralizer extension of a BP-subgroup of G[[r]] is BP and embeddable. Applications include embeddings of finitely generated fully residually free groups and their Lyndon completions, answering a question in [Br, Problem 4.15]. The paper also proves a real-coefficient analogue, Theorem 1.5.

Significance. If the proof is complete, the paper would answer a concrete open problem about embeddings of surface group fundamental groups into G[[r]] and would enlarge the known class of BP-subgroups of G[[r]]. The centralizer-extension part (Theorem 3.1(b) and Lemma 3.6) is coherent and appears sound, and the overall strategy is natural. However, the free-product case with trivial amalgam contains a substantial gap: the separation condition is applied to a word that is not known to lie in the BP-group H0. This gap affects the proof of Theorem 1.2 for chains involving trivial amalgams, and hence also Example 1.3(1). The application to fully residually free groups relies on centralizer extensions and may remain valid even if the trivial-amalgam case is not repaired.

major comments (2)

[Section 3.1, Lemma 3.5] The proof of the trivial-amalgam case invokes the separation condition of the BP-group H0 for the identity h1 c^{-n} h2 c^n ... h_{2k-1} c^{-n} h_{2k} c^n = 1 obtained after replacing the transcendental t by an integer n. But in this case c := exp(s e1 + s^2 e2) was chosen outside G_{σ(C)}[[r]], and H0 ≤ G_{σ(C)}[[r]] by the initial reduction; hence c^{±n} ∉ H0 for n ≠ 0. The word is therefore not known to lie in H0, and the separation condition cannot be applied to it. The reference to 'arguing as in the proof of Lemma 3.3' is not valid, because in Lemma 3.3 the element c belongs to H0 and the word lies in H0. Without the separation step, the word-reduction argument does not go through, so the isomorphism of Lemma 3.5 is unproved. The same problem affects the nontrivial-intersection alternative inside Lemma 3.4, where the separation condition is invoked for an equality involving the same element c outside H0.
[Section 1, Theorem 1.2 / Example 1.3(1)] Because the trivial-amalgam case of Theorem 3.1(a) is not established, the transfinite induction in Section 3.2 does not cover chains in which some C_alpha = {1}. In particular, Example 1.3(1), where G_{alpha+1} = G_alpha * G0 with trivial amalgam, is not proved. The 'if' direction of Theorem 1.2 therefore lacks support for an important class of admissible chains. The embedding of fully residually free groups (Example 1.3(2)) uses only centralizer extensions and may survive, but the statement of Theorem 1.2 should be restricted or the proof of Lemma 3.5 repaired before the main result can be accepted as stated.

minor comments (5)

[Abstract] The phrase 'of formal power se ries' contains a spacing typo; it should read 'of formal power series'.
[Section 2.2, formula (2.1)] In the displayed formula for v_r(1), the expression '(i1 + i2 + 1) · · · (i − ik + 1)' appears to have a missing parenthesis or a formatting error; please check the original computation.
[Lemma 3.6] The notation φ(1) for the image of the generator of the free factor Z is confusing, since 1 also denotes the identity element of G; using φ(z) or φ(t) would be clearer.
[Lemma 3.6, proof] The transition from (3.2) to 'g1 u^{α_1 n} ... gk u^{α_k n} = 1, n ∈ Z' is terse; an explicit sentence explaining that the identity is formal in the transcendental s, so one may substitute s = n, would improve readability.
[Section 3.2] The proof uses the fact that the union of a chain of BP-groups is BP; this is stated in the introduction without proof or reference, and a citation (or a short justification) would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the embedding theorem is proved constructively from external BP-group results and self-contained lemmas on G[[r]].

full rationale

The derivation chain is self-contained relative to the external theorems it imports. The 'if' direction of Theorem 1.2 is proved by transfinite induction using Theorem 3.1, which constructs embeddings of free products with amalgamation and centralizer extensions directly into G[[r]] via explicit generators such as c^{-s} H2 c^s with transcendental s. The inputs are: the BP/separation equivalence from KMS (Prop. 1), closure results for BP groups from KMS (Thm. 4, Cor. 6, Prop. 5), and the paper's own Lemmas 2.1 and Proposition 2.2, both proved from the explicit exponential/log formulas (2.1)-(2.3). The conjugating elements c = exp(se1 + s^2e2) are a constructive ansatz, not a renamed consequence of the theorem being proved. Embeddability of the final group is never used as an assumption except for the base G0; the 'if and only if' statement reduces the general case to the base case. The self-citation [Br, Problem 4.15] merely states the motivating question; it is not used as evidence for the embedding. The external citations [MR] and [KM] supply structure theorems for limit groups and Lyndon completions, and those results are independent of G[[r]]. Thus no step exhibits a reduction of the conclusion to its own inputs. (A referee note: the skeptical concern about Lemma 3.5 — whether the integer-parameter word lies in H0 when c is outside G_{sigma(C)}[[r]] — is a potential proof gap, not a circularity, and does not affect this score.)

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The paper imports BP-group theorems and the MR/KM structure theory of limit groups. The centralizer-extension part then carries the advertised consequence. The only fragile imported assumption is the BP separation condition itself, which the proof sometimes applies to words that may leave the ambient BP subgroup.

assumptions (6)

standard math ZFC set theory with the cardinality of the continuum and transcendence bases over Q.
Used to choose the embedding sigma and the injections tau_i in Section 3.2.
domain assumption The exponential and logarithm maps on G[[r]] are bijections with the stated coefficient formulas (2.1) and (2.2).
Standard structure of the prounipotent group G[[r]], used throughout the paper.
domain assumption Lemma 2.1: elements exp(a1) and exp(a2) commute if and only if a1 and a2 are proportional.
The paper proves this lemma, but it is a structural fact about the Lie algebra of formal vector fields that underpins the CSA property and all later arguments.
domain assumption [KMS, Thm. 4 and Cor. 6]: free products over centralizers and centralizer extensions of BP-groups are BP under suitable CSA conditions.
Used in Section 3.1 to conclude that the constructed groups are BP; the paper does not reprove these results.
domain assumption [MR] and [KM]: every finitely generated fully residually free group embeds into a finite sequence of centralizer extensions of the free group of rank two.
Used in Example 1.3(2) to obtain the advertised embedding of limit groups into G[[r]].
domain assumption [MR]: the Lyndon completion of a finitely generated fully residually free group is a direct limit of a countable chain of centralizer extensions.
Used in Example 1.3(3) to embed the Lyndon completion into G[[r]].

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Pith. "Pith review of Subgroups of the Group of Formal Power Series with the Big Powers Condition." pith.science (2026). https://pith.science/paper/PPAQWNZQ

@misc{pith2026190804918,
  author       = {Pith},
  title        = {Pith review of: Subgroups of the Group of Formal Power Series with the Big Powers Condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PPAQWNZQ}},
  note         = {Machine review of arXiv:1908.04918}
}

abstract

We study the structure of discrete subgroups of the group $G[[r]]$ of complex formal power series under the operation of composition of series. In particular, we prove that every finitely generated fully residually free group is embeddable to $G[[r]]$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Surface Groups In The Group Of Germs Of Analyticdiffeomorphisms In One Variable
math.GR 2019-09 conditional novelty 7.0 of 10

Surface groups admit faithful actions by germs of one-variable analytic diffeomorphisms, answering a question of Ghys.

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Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Baumslag, On generalised free products, Math

G. Baumslag, On generalised free products, Math. Z. 78 (1962), 423--438

work page 1962

[2] [2]

Baumslag, Residually free groups, Proc

B. Baumslag, Residually free groups, Proc. Lond. Math. Soc. 17 (1967), 402--418

work page 1967

[3] [3]

Brudnyi, Some algebraic aspects of the center problem for ordinary differential equations, Qual

A. Brudnyi, Some algebraic aspects of the center problem for ordinary differential equations, Qual. Theory Dyn. Syst. 9 (2010), 9--28

work page 2010

[4] [4]

Cerveau, Une liste de probl\`emes, in Ecuaciones Differenciales, Univ

D. Cerveau, Une liste de probl\`emes, in Ecuaciones Differenciales, Univ. Valladolid (1997), 455--460

work page 1997

[5] [5]

Cerveau and F

D. Cerveau and F. Loray, Un th\'eor\`eme de Frobenius singulier via l'arithm\'etique \'el\'ementaire, J. Number Theory 68 (1998), no. 2, 217--228

work page 1998

[6] [6]

Ecalle, B

J. Ecalle, B. Vallet, Intertwined mappings, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 3, 291--376

work page 2004

[7] [7]

Yu. S. Ilyashenko and A. S. Pyartli, The monodromy group at infinity of a generic polynomial vector field on the complex projective plane, Russian J. Math. Phys. 2 (1994), no. 3, 275--315

work page 1994

[8] [8]

Kharlampovich and A

O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group, II. Systems in triangular quasi-quadratic form and description of residually free groups, J. Algebra 200 (1998), no. 2, 517--570

work page 1998