The Higman--Thompson groups $V_n$ are $(2,2,2)$-generated

Eduard Schesler; Rachel Skipper; Xiaolei Wu

REVIEW 1 major objections 4 minor 2 cited by

The Higman--Thompson groups $V_n$ are $(2,2,2)$-generated

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Pith's one-line read The paper proves that for every n ≥ 2, the Higman–Thompson group V_n is generated by three involutions.

desk verdict A genuinely new and likely correct construction proving V_n is (2,2,2)-generated, but the final maximality step is an unproved generalization from V to V_n that a referee must check. read the letter →

arxiv 2411.09069 v1 pith:G57ATPY5 submitted 2024-11-13 math.GR

Eduard Schesler , Rachel Skipper , Xiaolei Wu This is my paper · ORCID

classification math.GR MSC 20F0520E32

keywords Higman–Thompsongroups(222)-generatedinvolutionsspinalelementscommutatorstablegeneratingsetsuniquedifferencepropertymaximalsubgroupsbranch

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 answers an open question by proving that every Higman–Thompson group V_n, for n ≥ 2, can be generated by three involutions. These groups act on the Cantor set of infinite words over an n-letter alphabet, and they are infinite simple groups (up to an index-2 subgroup when n is odd). The authors build, for each admissible sequence α, a triple {σ̂, τ, s_α}, where σ̂ swaps the first two letters, τ is a fixed involution with support near the left edge of the Cantor set, and s_α is a spinal involution, a homeomorphism acting along one thin branch of the tree of finite words. The proof shows this triple generates all of V_n, settling the (2,2,2)-generation case of that question.

What carries the argument

The spinal involution s_α is the central object: it is defined by s_α = $x_1^{{ℓ+1}}$ σ̂ ∏_{k=1}^{ℓ} x_1^k x_2 α_k and it is a genuine involution exactly when each α_k has order at most 2. The argument is carried by two structural properties of the sequence: the unique difference property, which makes the commutators [s_α, $s_α^{{t^k}}$] isolate single factors x_1^j x_2 [α_j, α_i] without overlap, and commutator stability, meaning the commutators [α_i, α_j] already generate $V_n^{1}$, which lets those isolated factors build up the whole derived subgroup. These lemmas construct larger and larger subgroups of V_n until the maximality of M forces equality.

What would settle it

Exhibit, for some n ≥ 2, a proper subgroup K of V_n satisfying ⊕_{i=1}^n x_i V_n ⋊ Sym(X_n) < K < V_n; this would invalidate Proposition 2.4 and remove the final step of the proof. In practice, one could compute the normalizer of the natural first-level subgroup in V_n for small n and check whether any element outside it generates a proper overgroup.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for every integer n ≥ 2, the Higman–Thompson group V_n admits a generating set consisting of three involutions. The discovery is a uniform construction rather than an existence argument alone: given a sequence α = (α_1, …, α_ℓ) of involutions in the derived subgroup $V_n^{1}$ which is commutator stable, with the positions of non-identity entries satisfying the unique difference property, the three involutions σ̂, τ, and s_α generate V_n. The proof establishes that this subgroup contains the natural subgroup M = ⊕_{i=1}^n x_i V_n ⋊ Sym(X_n), then uses maximality of M in V_n to conclude the subgroup is the whole group.

Load-bearing premise

The proof relies on the claim, imported from the n=2 case without a written proof, that the subgroup consisting of self-similar copies of V_n on the n top-level branches together with all permutations of those branches is a maximal subgroup of V_n.

Editorial extensions

If this is right

For every n ≥ 2, the group V_n is (2,2,2)-generated, answering the first half of Question 1.1 in the affirmative.
Since V_n is not (2,2)-generated by involutions, three is the minimal possible number of involutions in a generating set for V_n.
The generating triples are parametrized by admissible sequences α, giving a family of small generating sets rather than a single isolated example.
The proof connects Higman–Thompson groups to spinal-element techniques from branch group theory, suggesting a shared toolbox for generation questions.

Reading between the lines

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

The construction is uniform in n, but the admissible sequence α is guaranteed through existence arguments using finite generation and simplicity of V_n^1; for a fixed n, writing down an explicit α and the three involutions may require nontrivial computation.
The paper answers only the (2,2,2) half of Question 1.1; whether V_n is (2,3)-generated remains open, and nothing in this proof either supplies or rules out such a generating pair.
Because s_α is modelled on spinal tree automorphisms, the same three-involution scheme might extend to other groups acting on rooted trees, provided analogues of the maximality and simplicity lemmas hold.

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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

Colleague, here's my take on Schesler–Skipper–Wu. The headline result is real: they show every V_n is (2,2,2)-generated by three explicit involutions, improving the four-involution bound and answering the first part of CHMV Question 1.1. The construction is fresh: a spinal-type involution s_α plus the two standard elements σ and τ, with a unique-difference condition that makes commutators of s_α with its conjugates produce exactly the generators they need. The commutator algebra in Lemmas 3.8–3.11 is careful and checks out; the subdirect-product argument and use of local finiteness of the volume-preserving subgroup is clean. I found no circularity: they're not assuming the result.

The soft spot is exactly where the stress-test note lands. Proposition 2.4 — the maximality of ⊕ x_i V_n ⋊ Sym(X_n) in V_n — is the sole basis for the last line of Theorem 3.14, and it is stated as a 'same proof' generalization of [BBQS22, Prop 6.3], which is proved only for V (=V_2). The worry that the V-proof uses the fact that the complement of a first-level cone is a single cone, and that this breaks for n≥3, is a reasonable one, and I don't think a reader should have to take that on faith. This is load-bearing, not cosmetic. There's also a smaller issue: V_n^1 appears in Lemma 3.5 without definition; presumably it's the index-2 simple subgroup for odd n, but it needs to be spelled out.

That said, the maximality claim itself may well be true — the proof might carry over with minimal changes — but the authors should either provide the proof or cite a version that covers V_n. As written, a skeptical referee could not verify the main theorem without going to the source and doing the generalization themselves.

Who's this for? People studying generating sets of infinite simple groups; it's a concrete, citable advance that also introduces a transferable technique. If the maximality gap is patched, I'd want to cite it. I'd send it to peer review — the result is important enough and the construction solid enough to deserve referee time, and the referee can be sent to chase down Proposition 2.4. Not a desk reject.

Referee Report

1 major / 4 minor

Summary. The paper constructs, for each n ≥ 2 and each finite sequence α = (α_1, …, α_ℓ) of involutions in a certain derived subgroup V_n^1, a triple of involutions (9σ, τ, s_α) in the Higman–Thompson group V_n. The main theorem (Theorem 1.2) asserts that this triple generates V_n, thereby answering positively the question whether V_n is (2,2,2)-generated for every n. The proof shows that the subgroup H_α generated by the triple contains M := ⊕_{i=1}^n x_i V_n ⋊ 9Sym(X_n), and then uses the asserted maximality of M in V_n (Proposition 2.4) together with τ ∉ M to conclude H_α = V_n.

Significance. If the gap around Proposition 2.4 is repaired, this is a significant and clean result: it gives an explicit positive answer to a natural question of Corson–Hughes–Müller–Varghese and produces a whole family of three-involution generating sets for every Higman–Thompson group V_n. The core of the argument (Lemmas 3.8–3.13) is self-contained and correct: the commutator computations are transparent, the unique-difference property is used effectively to kill cross terms, the subdirect-product lemma is sound, and the construction is explicit rather than existential. The paper is also careful to make the generating involutions concrete and checkable.

major comments (1)

[§2, Proposition 2.4; §3, Theorem 3.14] The final step of the main proof is load-bearing and rests entirely on Proposition 2.4, which asserts that M := ⊕_{i=1}^n x_i V_n ⋊ 9Sym(X_n) is maximal in V_n. No proof is supplied; the text says only that it is a generalization of [BBQS22, Proposition 6.3] and that “the same proof given there, replacing V with V_n, works here as well.” This transfer is not automatic: [BBQS22] treats Thompson's group V = V_2, and for n ≥ 3 the complement of x_1 C_n is a union of n − 1 cones, whereas in V_2 it is a single cone. If any step in the maximality argument uses the two-cone structure, Proposition 2.4 may fail for n ≥ 3. Because Theorem 3.14 uses maximality as the sole mechanism to promote the inclusion M ≤ H_α to H_α = V_n, the authors must either include a proof of Proposition 2.4 for all n, or cite a source where it is proved for general V_n, and should explain why the multi-cone complement does not affect the argument. Without this, the proof of Theorem 1.2 is incomplete.

minor comments (4)

[§3, Lemma 3.5] The notation V_n^1 is used throughout but never defined; the reader must infer that it denotes the derived (or the finite-index simple) subgroup of V_n. Please define it explicitly, especially because when n is odd V_n itself is not simple.
[§2, Theorem 2.5] Theorem 2.5, that the abelianization of V_n is generated by the image of 9(x_1,x_2), is stated without proof or citation. It is used in Lemma 3.12 to pass from V_n^1 and 9σ to V_n, so a reference (for example to Higman's book) or a short proof should be added.
[§3, Definition 3.1] The case analysis in the displayed definition of s_α appears to omit the case k = 0, i = 1. Formula (3.1) makes the intended definition clear, but the displayed definition should be rewritten as a partition of C_n so that the domain is unambiguous.
[References] The reference [CHMV23] is given with a Google Drive link; a stable arXiv identifier or a journal reference would be more appropriate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is an explicit commutator construction; Proposition 2.4 is a deferred maximality lemma and a proof-gap risk, not a reduction to the target.

full rationale

The derivation of Theorem 1.2 is self-contained on its constructive side. S_α = {9σ, τ, s_α} is defined independently of the conclusion; Lemmas 3.8–3.12 use commutator identities, the unique-difference property, commutator-stable generating sets of the simple finitely generated derived subgroup V_n', and Higman's computation of V_n^ab to show H_α contains the copies x_1 V_n and then x_i V_n, and Lemma 3.13 obtains 9Sym(X_n). No parameter is fitted and no conclusion is assumed. The final step invokes Proposition 2.4, the maximality of ⊕_{i=1}^n x_i V_n ⋊ 9Sym(X_n) in V_n; the paper states 'The following is a generalization of [BBQS22, Proposition 6.3]. The same proof given there, replacing V with V_n, works here as well.' Since [BBQS22] is about V=V_2 and has an overlapping author, the unproved generalization is a real correctness gap for n≥3, and Theorem 3.14 is incomplete as written. However, maximality is a distinct mathematical fact, not the (2,2,2)-generation property itself, and the inference H_α=V_n from a maximal subgroup plus an outside element is a standard lattice argument. The dependency is a deferred external lemma, not a definitional or fitted reduction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard theorems about V_n (simplicity of the derived subgroup, abelianization) and on a maximal subgroup statement generalized from [BBQS22] without proof. The construction of α involves no fitted numbers; the 'unique difference property' set is chosen combinatorially. No entities are invented.

assumptions (3)

domain assumption The derived subgroup V_n^1 of V_n is simple for all n≥2.
Used in Lemma 3.5 to build a commutator-stable generating set of involutions; cited to Higman [Hig74, Theorem 5.4].
domain assumption The abelianization V_n^{ab} is generated by the image of 9σ.
Theorem 2.5; used in Lemma 3.12 to pass from V_n^1 to V_n.
domain assumption The subgroup M = ⊕_{i=1}^n x_i V_n ⋊ 9Sym(X_n) is maximal in V_n.
Proposition 2.4; asserted as generalization of [BBQS22, Proposition 6.3] with no proof given; load-bearing in Theorem 3.14.

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Pith. "Pith review of The Higman--Thompson groups $V_n$ are $(2,2,2)$-generated." pith.science (2026). https://pith.science/paper/G57ATPY5

@misc{pith2026241109069,
  author       = {Pith},
  title        = {Pith review of: The Higman--Thompson groups $V_n$ are $(2,2,2)$-generated},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G57ATPY5}},
  note         = {Machine review of arXiv:2411.09069}
}

abstract

We provide a family of generating sets $S_{\alpha}$ of the Higman--Thompson groups $V_n$ that are parametrized by certain sequences $\alpha$ of elements in $V_n$. These generating sets consist of $3$ involutions $\sigma$, $\tau$, and $s_{\alpha}$, where the latter involution is inspired by the class of spinal elements in the theory of branch groups. In particular this shows the existence of generating sets of $V_n$ that consist of $3$ involutions.

Figures

Figures reproduced from arXiv: 2411.09069 by the authors.

**Figure 1.** The action of σ9 P V5 on C5. Let τ P Vn denote the unique element with supppτ q Ď ptx1u ˆ pXnztxnuqqCn Y pXnztx1uqCn that satisfies τ px1xiξq “ xi`1ξ and τ pxi`1ξq “ x1xiξ for all 1 ď i ă n and ξ P Cn [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** The action of τ P V5 on C5. The generating set Sα of Vn that we want to construct consists of the elements σ, τ 9 and one further involution sα, where α is a sequence in V ˚ n :“ š ℓPN0 V ℓ n . The length of a sequence α “ pα1, . . . , αℓq P V ˚ n will be denoted by |α| “ ℓ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The action of sα P V3 on C3, where α “ pα1, α2, α3q. Using Notation 2.3, we can write sα as (3.1) sα “ x ℓ`1 1 σ9 ¨ ź ℓ k“1 x k 1x2αk. Note that sα is an involution if and only if α “ pαiq ℓ i“1 consists of elements αi of order at most 2. Our goal is to find conditions on α under which the group Hα generated by Sα :“ tσ, τ, s 9 αu coincides with Vn. Remark 3.2. The definition of sα is inspired by the class of tree a… view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

16 extracted references · 14 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Type systems and maximal subgroups of Thompson's group $V$

James Belk, Collin Bleak, Martyn Quick, and Rachel Skipper. Type systems and maximal subgroups of T hompson's group V . Trans. Amer. Math. Soc. , 2022. to appear: arXiv:2206.12631

work page Pith review arXiv 2022

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Brown and Ross Geoghegan

Kenneth S. Brown and Ross Geoghegan. An infinite-dimensional torsion-free FP _ group. Invent. Math. , 77(2):367--381, 1984

work page 1984

[3] [3]

Grigorchuk, and Zoran S uni\' k

Laurent Bartholdi, Rostislav I. Grigorchuk, and Zoran S uni\' k . Branch groups. In Handbook of algebra, V ol. 3 , volume 3 of Handb. Algebr. , pages 989--1112. Elsevier/North-Holland, Amsterdam, 2003

work page 2003

[4] [4]

Finitely presented simple groups and products of trees

Marc Burger and Shahar Mozes. Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris S\'er. I Math. , 324(7):747--752, 1997

work page 1997

[5] [5]

The infinite simple group V of R ichard J

Collin Bleak and Martyn Quick. The infinite simple group V of R ichard J . T hompson: presentations by permutations. Groups Geom. Dyn. , 11(4):1401--1436, 2017

work page 2017

[6] [6]

Kenneth S. Brown. Finiteness properties of groups. In Proceedings of the N orthwestern conference on cohomology of groups ( E vanston, I ll., 1985) , volume 44, pages 45--75, 1987

1985

[7] [7]

J. W. Cannon, W. J. Floyd, and W. R. Parry. Introductory notes on R ichard T hompson's groups. Enseign. Math. (2) , 42(3-4):215--256, 1996

1996

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Corson, Sam Hughes, Philip M\"oller, and Olga Varghese

Samuel M. Corson, Sam Hughes, Philip M\"oller, and Olga Varghese. Higman-thompson groups and profinite properties of right-angled coxeter groups. arXiv preprint arXiv:2309.06213. More recent version available at https://drive.google.com/file/d/100swo52dOxiRTuK-xeRVD1eJMntWRozg/view , 2023

work page arXiv 2023