REVIEW 2 major objections 4 minor
Generating simple vigorous groups
T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read Every finitely generated simple vigorous group is (m,n)-generated for every m≥2 and n≥3, and Thompson's group V is (2,3)-generated.
desk verdict Strong results and a likely-correct main theorem, but Proposition 3.3(vi) has a real gap that needs fixing before the proof is sound. 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 load-bearing device is a two-stage construction. First, a previously established technical result guarantees that every finitely generated simple vigorous group has a generating pair (ρ,σ) with controlled orders and a noncommutativity condition [σ^k,(ρ^εσ)^k]≠1. Second, a transfer map Φ_{m,n} builds from such a pair—living on a small clopen set—two global elements α,β of orders m and n. The transfer is carried by encoded permutations on marked clopen tiles, and its key algebraic output is that commutators of certain words in α,β recover the commutators supplied by the input pair. Along with the vigor-based small-support combination lemma, which joins subgroups supported on overlapping cl
What would settle it
Run the paper's explicit construction for a concrete finitely generated simple vigorous group—for instance, check the displayed pair for Thompson's group V and verify directly that the two elements have orders 2 and 3 and generate the group; any failure would refute Theorem 1. A broader test is to find any finitely generated simple vigorous group for which the cited input-pair guarantee fails to provide a pair with the stated orders and commutator condition.
Extended reading notes
Core claim
On the authors' terms: every finitely generated simple vigorous group G admits, for each m≥2 and n≥3, a generating pair (α,β) with |α|=m and |β|=n; moreover the order-n generator can be arranged to be a product of two involutions, so G is generated by three involutions. The construction yields infinitely many pairwise inequivalent strongly minimal generating sequences, minimal generating sets of every size k≥2 with prescribed orders, and a uniform-spread witness: a single σ∈G of order 30 such that every nontrivial element pairs with some conjugate of σ to generate G. For Thompson's group V the paper writes down an explicit (2,3)-generating pair, and it derives from the general theorem that e
Load-bearing premise
The whole argument rests on the claim that every finitely generated simple vigorous group supplies a generating pair whose powers interact nontrivially in a specific iterated commutator way; if even one such group lacked such a pair, the entire construction would lack its starting material.
Editorial extensions
If this is right
- Thompson's group V is (2,3)-generated: two elements of orders 2 and 3 generate it, resolving the previously open question.
- Every finitely generated simple vigorous group is generated by three involutions; for V the paper displays three very short involutions.
- Every finitely generated group quasi-isometrically embeds in a (2,3)-generated simple group, for every allowed pair (m,n).
- Every such group has minimal generating sets of every size k≥2 with prescribed element orders, so there is no upper bound on the size of an irredundant generating set.
- The uniform spread of every such group is at least 1: one fixed element of order 30 pairs with a conjugate to generate the group through any specified nontrivial element.
Reading between the lines
- If the input-pair theorem extends beyond vigorous groups to other classes of infinite simple groups, the same transfer construction would likely yield (m,n)-generation for those classes as well, making the method a template for infinite simple group generation problems.
- The explicit (2,3)-generating pair for V suggests that the uniform spread of V may be infinite; a natural next step is to test whether the uniform-spread witness of order 30 can be replaced by a single element that works for arbitrarily many nontrivial elements simultaneously.
- Because the construction is so flexible, it might be possible to combine it with existing embeddings into vigorous groups to obtain (m,n)-generated simple groups with additional properties, such as solvable word problem or finite presentation when the ambient group has them.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves several strong generation results for finitely generated simple vigorous groups, a class of homeomorphism groups of Cantor space that includes Thompson's group V and many of its generalizations. The main results are: (1) for every m ≥ 2 and n ≥ 3, every such group G is (m,n)-generated; in particular V is (2,3)-generated, answering a question of Sapir; (2) every such group is generated by three involutions; (3) it has minimal generating sets of every size k ≥ 2, and more generally of specified element orders; (4) it satisfies a strong form of 3/2-generation, with uniform spread at least 1; and (5) direct powers of such groups are 2-generated. The proofs are constructive, with explicit generators given for V (e.g., Example 3 and Corollary 3.7). A general generation criterion (Proposition 3.3) is the technical heart, converting a generating pair of a rigid stabilizer into a generating pair of the whole group with prescribed orders.
Significance. If the main results are correct, they constitute a substantial advance in the theory of infinite simple groups. They transfer several well-known properties of finite simple groups — (m,n)-generation, generation by three involutions, 3/2-generation, and uniform spread — to a broad infinite family, and they answer Sapir's question about V. The paper also provides new tools, such as the generation criterion Theorem 2.9 and the general construction of Proposition 3.3, which are likely to have further applications. A notable strength is the constructive nature of the arguments, with explicit generators for V in several cases. The main issue is a false equality in the proof of Proposition 3.3(vi), which is load-bearing for all of the paper's central claims; however, the flaw appears local and plausibly repairable, so the underlying results are likely sound.
major comments (2)
- [§3.1, Proposition 3.3(vi), around Eq. (3.5)] The proof that G_C ≤ ⟨α,β⟩ rests on the assertion S={([σ^r,τ^r])^θ | θ∈⟨α,β⟩}={([σ^r,τ^r])^θ | θ∈G_C}. This equality is false. Since β maps C to Cγ_{01} (disjoint from C) and a=[σ^r,τ^r] has support in C, the conjugate a^β has support Cγ_{01} and does not belong to G_C. The right-hand set consists of elements supported in C, so the claimed equality fails. Consequently the conclusion ⟨S⟩=G_C, and hence G_C≤⟨α,β⟩, is unsupported. The argument can likely be repaired by defining S={a^h | h∈H}, where H is the setwise stabiliser of C in ⟨α,β⟩; then (3.2) implies S is a nontrivial normal subset of the simple group G_C, so ⟨S⟩=G_C, and a∈⟨α,β⟩ gives S⊆⟨α,β⟩. However, as written the proof is incorrect, and this step is load-bearing for Theorem 1 and all subsequent results.
- [§3.2, Proposition 3.4] The proof of Proposition 3.4(ii) uses the same flawed equality in the form ⟨w_i^{⟨u_i,v_i⟩}⟩=G_C. The same repair applies if one replaces ⟨u_i,v_i⟩ by the setwise stabiliser of C. Since Proposition 3.4(ii) is used in Theorem 3.5 to produce infinitely many Aut(G)-inequivalent generating sequences, this should be corrected consistently with the repair of Proposition 3.3(vi).
minor comments (4)
- [§2.3, Proposition 2.8] Please make explicit that Proposition 2.8 is [4, Theorem 5.15] and that the proof given here is only an outline; as written it could be mistaken for a new self-contained proof. A precise reference to the published result would avoid any ambiguity.
- [Example 3 and Corollary 3.7] The notation γ[101] and γ[111] is introduced in Example 3, but the definitions rely on elements ϕ,ψ that are only defined later in the example. It would be helpful to define ϕ and ψ before using them, or to flag that the definitions appear below.
- [Example 8] There is a typo: 'let x1,...,xN be the of words' should read 'let x1,...,xN be the words'.
- [Author affiliations] The third author's affiliation contains a typo: 'University of Birminhgham' should be 'University of Birmingham'.
Circularity Check
No significant circularity: the paper's new constructions are not equivalent to their inputs, although one written reduction step in Proposition 3.3(vi) appears incorrect.
full rationale
The derivation chain is anchored in Proposition 2.8, imported from [4] (Bleak–Elliott–Hyde, with two of the current authors). This is a load-bearing element of the paper, and the author overlap is real. But it is a published, parameter-free theorem whose hypotheses (finitely generated simple vigorous) do not contain the target conclusions (exact (m,n)-generation, three involutions, minimal generating sets of every size >=2, uniform spread >=1). The actual contribution of this paper is the new Phi construction (Definition 3.2) and the estimates in Proposition 3.3, Theorem 2.9, and Section 4; these genuinely convert the input pairs into stronger statements rather than renaming them. The use of [12, Lemma 3.4(i)] in Theorem 2.9 is likewise an external lemma with no content overlap. The explicit V examples are verified by direct computation, not fitted. The one point that warranted scrutiny is the equality in Proposition 3.3(vi): S={[sigma^r,tau^r]^theta | theta in <alpha,beta>}={[sigma^r,tau^r]^theta | theta in G_C}. As written this appears false, since beta does not setwise stabilise C, so conjugating by beta can move the support of [sigma^r,tau^r] off C and outside G_C. This is a serious correctness gap in the written proof of G_C <= <alpha,beta>, but it is not a circularity: it is an unjustified equality in a reduction, not an identification of the target theorem with the input hypothesis. The natural repair (conjugating by the setwise stabiliser of C) would make the step a legitimate simplicity argument. Therefore the circularity score is 0; the proof-gap concern belongs to a correctness review, not to the circularity pass.
Assumptions & free parameters
assumptions (3)
- standard math ZFC set theory and standard group theory
- domain assumption The structural theory of vigorous and approximately full groups from [4]: Lemmas 2.1–2.5, the homology group X_G, and Proposition 2.8 (existence of ρ,σ with prescribed orders and the commutator nonvanishing condition)
- domain assumption Existence of a proper clopen set C⊊Cantor with [C]_G even (used in Definition 3.2 and Theorem 3.5), and [D]_G=-2[C]_G in Proposition 2.8
Cite this review
Pith. "Pith review of Generating simple vigorous groups." pith.science (2026). https://pith.science/paper/PCCLASUL
@misc{pith2026260721754,
author = {Pith},
title = {Pith review of: Generating simple vigorous groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/PCCLASUL}},
note = {Machine review of arXiv:2607.21754}
}
abstract
The simple vigorous groups form a broad class of groups of homeomorphisms of Cantor space that includes Thompson's group $V$, its various generalisations and many others such as Nekrashevych's groups of dynamical origin. Bleak, Elliott and Hyde (2024) proved that every finitely generated simple vigorous group is $2$-generated, and, in this paper, we give several strong generation results for this class of simple groups. For example, we prove that if $G$ is a finitely generated simple vigorous group, then $G$ is generated by three involutions, $G$ is generated by an element of order $m$ and an element of order $n$ for any choice of $m \geq 2$ and $n \geq 3$, $G$ has a minimal generating set of size $k$ for all $k \geq 2$, every nontrivial element of $G$ is contained in a generating pair and the direct power $G^n$ is $2$-generated for all $n$. These results are analogous to well-known results for finite simple groups, but of course the proofs in this context are quite different. One consequence of our results is that Thompson's group $V$ is $(2, 3)$-generated, which answers a question of Sapir (2017). Another consequence is that every finitely generated group quasi-isometrically embeds in a $(2, 3)$-generated simple group, strengthening theorems of Hall (1974) and Bridson (1998). All of our proofs are constructive, and we establish several new generation criteria for these groups, which we expect to be of wider interest, even just for Thompson's group $V$.
Figures
Reviewed August 1, 2026 · model on record in the stance chip above.
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