Virtual First Betti Number of GGS Groups

Andrew Ng

REVIEW 2 major objections 4 minor 21 references

Virtual First Betti Number of GGS Groups

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read GGS group construction yields torsion-free, residually finite groups that are not virtually diffuse, answering a question from the diffuse-groups literature.

desk verdict Clean criterion and a real answer to an open question, but the p=3 case contradicts the paper's own Remark 14 and needs to be resolved before the theorem can be trusted. read the letter →

arxiv 2505.23269 v1 pith:WVTB7J7H submitted 2025-05-29 math.GR

Andrew Ng This is my paper

classification math.GR MSC 20F6520E08

keywords virtualfirstBettinumberGGSgroupsdiffuseuniqueproductpropertybranchamenabletorsion-freeresiduallyfinite

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 proves that for every odd prime p there is a torsion-free, finitely generated, residually finite group which is not virtually diffuse, while all of its finite quotients are p-groups. The existence of such groups settles a question from the diffuse-groups literature. The proof works with the derived subgroup of a Grigorchuk-Gupta-Sidki (GGS) group acting on a rooted p-ary tree, using the fact that it is a regular branch group: at every level of the tree it contains a copy of itself times itself, which forces an infinite direct sum of copies of the group to embed inside it. Because the group is amenable, a theorem relating diffuseness to local indicability turns the absence of homomorphisms to Z from finite-index subgroups into failure of virtual diffuseness.

What carries the argument

The load-bearing object is the class of Grigorchuk-Gupta-Sidki (GGS) groups: automorphism groups of the rooted p-ary tree generated by a cyclic permutation a of the p subtrees and a recursively defined element b. For the defining vector (1,...,1,lambda), lambda not 1 or 2, a cited theorem makes the derived subgroup K a torsion-free regular branch subgroup; regular branch means K contains, in its action on the first level, a copy of K times K, and the iteration of this fact embeds an infinite direct sum of copies of K. Two standard inputs complete the mechanism: bounded automata groups are amenable, so K is amenable, and for amenable groups diffuseness is equivalent to local indicability; Lemma 4, that a finitely generated group whose finite quotients all have order prime to a fixed prime has vanishing virtual first Betti number, supplies the obstruction to local indicability through finite-index subgroups.

What would settle it

For the smallest new case, p=3 with defining vector (1,2), compute the finite quotients of the derived subgroup: if any quotient has order divisible by a prime other than 3, or if any finite-index subgroup of the derived subgroup admits a nonzero homomorphism to Z, the theorem's claim for that group is false. Alternatively, inspect the first few levels of the rooted tree and check whether each level truly contains a copy of the derived subgroup times itself; if the embedded infinite direct sum cannot be found, the branch-structure step fails.

Watch

Extended reading notes

Core claim

The central discovery is that the torsion-free derived subgroup K of the GGS group with defining vector (1,...,1,lambda), for lambda not 1 or 2 and p odd, is a group of exactly the kind asked for: K is finitely generated and residually finite, all its finite quotients have order a power of p, and K is not virtually diffuse. The paper shows this in two ways: directly from the branch structure, where K embeds an infinite direct sum of copies of itself and this direct sum is shown not to be virtually diffuse; and via the congruence subgroup property, where the p-power order of congruence quotients forces vanishing virtual first Betti number, which for amenable groups rules out virtual diffuseness. This answers the question whether every torsion-free finitely generated residually finite group is virtually diffuse in the negative.

Load-bearing premise

The entire construction depends on the cited theorem that the GGS group with defining vector (1,...,1,lambda), lambda not 1 or 2, is just infinite and regular branch over its torsion-free derived subgroup; if that theorem fails for any of the chosen vectors, or if the derived subgroup is not finitely generated, the proof does not go through.

Editorial extensions

If this is right

The paper settles the diffuse-groups open question in the negative: torsion-free, finitely generated, residually finite groups need not be virtually diffuse.
All constructed groups have finite quotients of p-power order, so this class of groups is not forced to be virtually diffuse by its finite quotient structure.
The vanishing-virtual-Betti-number criterion supplies a reusable obstruction: an amenable finitely generated group with p-power finite quotients cannot be virtually diffuse.
The examples do not provide counterexamples to Baer's conjecture: GGS groups are branch groups and therefore not Noetherian, so their group rings are not Noetherian.

Reading between the lines

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

If the Lemma 4 criterion is combined with the amenable-group equivalence, any infinite finitely generated amenable group whose finite quotients all have order prime to a fixed prime should fail to be virtually diffuse; the GGS examples are only the first concrete instances.
The paper's open question about the unique product property suggests a testable next step: compute whether the derived subgroup K has unique products; if it does, K would be a unique-product group that is not diffuse, a combination not previously exhibited.
The first proof avoids the congruence subgroup property entirely, so variants of the construction that lack CSP but retain amenability and branch structure should also yield non-virtually-diffuse groups; this could be checked in other automata groups with p-power quotients.

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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

Short version: this is a nice short paper with a clean criterion, but as written it contradicts itself on the p=3 case. That has to be fixed before I'd trust Theorem 3.

The genuinely new thing is Theorem 3, which answers Kionke–Raimbault: infinitely many torsion-free, finitely generated, residually finite groups that aren't virtually diffuse, with finite quotients all p-groups. Lemma 4 is a simple and useful observation, and Proposition 13 looks correct. The first proof's strategy—use branch structure to embed an infinite direct sum of a non-diffuse amenable subgroup—is sensible. I also appreciate the second proof via the congruence subgroup property.

But the stress-test is spot on. Remark 14 explicitly says the generalised Fabrykowski–Gupta groups with vector (1,0,...,0), d ≥ 3, are not virtually torsion-free and their commutator subgroups aren't torsion-free. For p=3, the vector (1,0) is exactly the d=3 case, and that is the group used in Theorem 3. So the theorem's p=3 claim directly contradicts the remark (and the cited FGU17 Proposition 4.1, assuming the remark quotes it correctly). Either the theorem needs a different p=3 example with a proof that doesn't rely on the derived subgroup being torsion-free, or the remark has a typo and should say d ≥ 4. As written, this is a load-bearing gap: if p=3 is not covered, the 'for each odd prime' statement is false.

Other soft spots are minor. The 'iterate the construction' step embedding an infinite direct sum is sketched tersely; standard branch-group facts probably fill it, but it deserves a few more lines. The heavy lifting is external: FGU17 Theorems A and D, and the finite generation of the derived subgroup is not explicitly justified. I found no circularity and no fitted parameters.

Who this is for: anyone working on diffuse groups, the Kaplansky unit conjecture, or branch groups. The core idea is worth refereeing, but not in its current state. I'd send it to a referee with a clear instruction to pin down the p=3 case and clean up Remark 14. If the author can repair or remove the contradiction, this becomes a solid short paper.

Referee Report

2 major / 4 minor

Summary. The paper proves a criterion (Lemma 4) for vanishing virtual first Betti number and uses it, together with known results on GGS groups, to construct for each odd prime p a torsion-free, finitely generated, residually finite group that is not virtually diffuse. The constructed group is the derived subgroup of a GGS group with defining vector (1,...,1,λ), λ≠1,2. Two proofs are given: the first uses branch structure and a proposition about infinite direct sums of non-diffuse groups; the second uses the congruence subgroup property and explicit computation of congruence quotients. The paper also discusses connections to the Kaplansky unit conjecture and Baer's conjecture.

Significance. If correct, the main result answers in the affirmative a question of Kionke and Raimbault on the existence of torsion-free, finitely generated, residually finite groups that are not virtually diffuse. The construction is explicit and the paper is concise. The internal lemmas (Lemma 4 and Proposition 13) are clear and appear correct, and the proof transparently relies on well-established external results on GGS groups. The paper also gives two independent routes to non-virtual diffuseness, one of which avoids the congruence subgroup property.

major comments (2)

[§4, Remark 14 vs. proof of Theorem 3] There is a direct internal contradiction concerning the p=3 case. For p=3, the only admissible value λ≠1,2 is λ=0, giving the Fabrykowski-Gupta group with vector (1,0). The first proof of Theorem 3 claims, citing [FGU17, Theorem D], that the derived subgroup K of this group is torsion-free. However, Remark 14 states that the generalized Fabrykowski-Gupta groups, defined by the vector (1,0,...,0) for any integer d≥3, have commutator subgroups that are not torsion-free, citing [FGU17, Proposition 4.1]. Since d=3 gives exactly the same group as the p=3 vector (1,0) case, the two statements contradict each other. This is load-bearing because Theorem 3 asserts 'for each odd prime p', including p=3. The author must resolve this inconsistency: either Remark 14's range d≥3 should be corrected to d≥4 (if that is what the cited proposition actually says), or Theorem 3 must exclude p=3 or be given a separate argument for p=3. As written, the proof does not cover all odd primes.
[§4, first proof of Theorem 3] The branch-structure step that embeds an infinite direct sum of copies of K into G is only sketched. The text says that selecting a copy of K×K at each level yields copies of K that 'generate their infinite direct sum', but it does not justify that the selected subgroups commute and that their product is indeed an internal direct sum. Since this embedding is essential for applying Proposition 13, the argument should be spelled out more explicitly or a precise reference to the standard regular branch group construction should be given.

minor comments (4)

[§4, first proof of Theorem 3] The phrase 'so creflidiﬀ then applies' contains a garbled reference; it should presumably refer to [LW14, Theorem 6.4] (Theorem 7 of the paper).
[§4, first proof of Theorem 3] The sentence 'Suppose the branching subgroup K isn't diffuse, equivalently locally indicable' is confusing because the equivalence between diffuseness and local indicability holds only for amenable groups, and the statement is not an assumption but a conclusion derived from the finite abelianisation of K. The logical flow should be reworded.
[§4, Remark 14] The phrase 'at least when p ≥ 7 is a prime' is unclear because p has not been defined in the context of the generalized Fabrykowski-Gupta groups (where the degree is denoted d). It should be corrected to refer to d or be clarified.
[§4, second proof of Theorem 3] When applying Lemma 4 in the second proof, the text should explicitly state that q is chosen to be a prime different from p, since the finite quotients have order a power of p and are therefore prime to q for any such q.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's derivation is self-contained or relies on independent external theorems, with no fitted inputs or self-citations carrying the argument.

full rationale

The paper contains no fitted parameters, post-hoc exclusions, or author self-citations that carry the argument. Lemma 4 is proved directly: if a finite-index subgroup H of G surjected onto Z, then composing with reduction mod q would produce a finite quotient of G of order divisible by q, contradicting the assumption that all finite quotients have order prime to q. This is a genuine first-principles observation, not an input disguised as a conclusion. The main construction uses external theorems by other authors: [FGU17, Theorem D] supplies that the relevant GGS groups are just infinite and branch over a torsion-free derived subgroup for lambda not 1 or 2; [BGS05, Theorem 5.2] gives finite abelianisation of the derived subgroup; [BKN10] and [Ne05] give amenability of bounded automata groups; [LW14, Theorem 6.4] gives the equivalence between diffuse and locally indicable for amenable groups; [FZ14, Theorem A] computes congruence quotient orders as powers of p. Each of these is an independent, externally established result and none is authored by the present paper's author, so there is no self-citation chain. Proposition 13, the new embedding argument for the infinite direct sum, is proved in the text and does not assume the conclusion. The only notable concern in the manuscript is internal consistency of Remark 14 with the p=3 case, since the remark says generalised Fabrykowski-Gupta groups are not virtually torsion-free and their commutator subgroups are not torsion-free, while Theorem 3's p=3 example uses the derived subgroup of the Fabrykowski-Gupta group. That issue, if real, would undermine correctness of part of Theorem 3, but it is not a circularity: the theorem would fail or need repair rather than reduce to its own assumptions. No equation is postulated as both premise and conclusion, and no 'prediction' is a renamed fit. Therefore the circularity burden is not met, and the appropriate score is 0.

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

The paper introduces no free parameters or new entities. It consumes several published theorems as external inputs, all listed above.

assumptions (5)

domain assumption For lambda != 1,2, the GGS group G defined by vector (1,...,1,lambda) is just infinite and regular branch over its derived subgroup, which is torsion-free.
From FGU17, Theorem D, used in both proofs of Theorem 3 to identify the torsion-free subgroup K = G'.
domain assumption The derived subgroup of these GGS groups has finite abelianization (hence is not locally indicable).
From BGS05, Theorem 5.2; used to conclude K is not diffuse via LW14.
domain assumption An amenable group is locally indicable if and only if it is diffuse (LW14 Theorem 6.4).
Bridges finite abelianization to non-diffuseness for the amenable GGS groups.
domain assumption Bounded automata groups are amenable (BKN10, Ne05).
Used to show the GGS groups considered are amenable.
domain assumption The congruence quotients of GGS groups all have p-power order (FZ14 Theorem A), and GGS groups have the congruence subgroup property (FGU17 Theorem A), which passes to finite index subgroups.
Used in the second proof to show all finite quotients of the derived subgroup are p-groups, giving vanishing virtual first Betti number via Lemma 4.

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Cite this review

Pith. "Pith review of Virtual First Betti Number of GGS Groups." pith.science (2026). https://pith.science/paper/WVTB7J7H

@misc{pith2026250523269,
  author       = {Pith},
  title        = {Pith review of: Virtual First Betti Number of GGS Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WVTB7J7H}},
  note         = {Machine review of arXiv:2505.23269}
}

read the original abstract

We observe a criterion for groups to have vanishing virtual first Betti number and use it to give infinitely many examples of torsion-free, finitely generated, residually finite groups which aren't virtually diffuse. This answers a question raised by Kionke and Raimbault.

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Reference graph

Works this paper leans on

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Pith tools

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

[1] [1]

Bartholdi, B

L. Bartholdi, B. Eick and R. Hartung, A nilpotent quotient algorithm for certain infinitely presented groups and its applications, Internat. J. Algebra Comput. 18 (2008), no. 8, 1321--1344; MR2483125

work page 2008

[2] [2]

Bartholdi and R

L. Bartholdi and R. I. Grigorchuk, On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math. J. 28:1 (2002), 47–90

work page 2002

[3] [3]

Bartholdi, R

L. Bartholdi, R. I. Grigorchuk and Z. Suni\'c, Branch groups, in Handbook of algebra, Vol. 3 , 989--1112, Handb. Algebr., 3, Elsevier/North-Holland, Amsterdam, ; MR2035113

work page

[4] [4]

V. A. Belyaev, R. I. Grigorchuk, and P. Shumyatsky, On just-infiniteness of locally finite groups and their C^* -algebras, Bull. Math. Sci. 7 (2017), no. 1, 167–175

work page 2017

[5] [5]

Bartholdi, V

L. Bartholdi, V. A. Kaimanovich and V. V. Nekrashevych, On amenability of automata groups, Duke Math. J. 154 (2010), no. 3, 575--598; MR2730578

work page 2010

[6] [6]

B. H. Bowditch, A variation on the unique product property. J. London Math. Soc. (2)62(2000), no.3, 813–826

work page 2000

[7] [7]

Delzant, `Sur l'anneau d'un groupe hyperbolique', C

T. Delzant, `Sur l'anneau d'un groupe hyperbolique', C. R. Acad. Sci. Paris SeTr. I 324 (1997) 381–384

work page 1997

[8] [8]

S. P. Fisher, Improved algebraic fibrings, Compos. Math. 160 (2024), no. 9, 2203--2227; MR4797112

work page 2024