{"id":"4b4a0ae4-0969-4926-b95b-5f36229e6932","arxiv_id":"2607.21754","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every finitely generated simple vigorous group is (m,n)-generated for all m≥2, n≥3; in particular, Thompson's group V is (2,3)-generated.","lead":"Every finitely generated simple 'vigorous' group — a broad class including Thompson's group V — is shown to be generated by two elements with any prescribed finite orders, by three involutions, and to have minimal generating sets of every size. This answers Sapir's 2017 question: Thompson's group V is (2,3)-generated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.3(vi) proof defines S as conjugates of a nontrivial commutator by all of ⟨α,β⟩ and then asserts S equals its conjugates by G_C; because β maps C off itself, the equality is false as written, so the proof that G_C≤⟨α,β⟩ fails.","rationale":"The reader's weakest-assumption pick (Proposition 2.8, imported from [4]) is a genuine external dependency, but the more acute vulnerability is internal: the proof of Proposition 3.3(vi) contains an unjustified equality that appears false on its face because β moves the chosen clopen C off itself. Since Proposition 3.3 is the engine for Theorems 1, 4, 7, 9, 11, and their corollaries, a gap here undermines the paper's central claim. However, the flaw looks like it may be a simple notational slip (replacing H by ⟨α,β⟩); if the authors confirm that S is defined using the setwise stabiliser H, the argument may be recoverable. Thus I do not recommend outright rejection, but the paper should be accepted only conditionally on this point being fixed and the revised proof verified. I partially agree with the reader because they flagged an external dependency rather than this internal proof step.","tokens_in":26653,"tokens_out":17010,"duration_ms":147269,"concrete_test":"Pin down the intended definition of S in Proposition 3.3(vi). Take a concrete simple vigorous group, e.g. V with C=010C and σ,τ as in Corollary 3.7, and compute the Φ_{2,3} output (α,β). Let a=[σ^{112},τ^{112}]. Directly test whether a^β lies in V_C by checking whether its support is contained in C. According to Definition 3.2, β sends C to Cγ_{01} with C∩Cγ_{01}=∅, so a^β should have support disjoint from C, confirming the written equality S={a^θ | θ∈G_C} is false. Then re-run the proof of (3.5) with S={a^θ | θ∈H} (H the setwise stabiliser of C) to see whether the normality/simplicity argument yields G_C≤⟨α,β⟩; if yes, the manuscript has a repairable typo; if no, the main generation theorem is unproven.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central generation argument depends on Proposition 3.3(vi), whose proof claims that for a=[σ^r,τ^r] (with (σ,τ)∈S_{m,r}(G_C)), S={a^θ | θ∈⟨α,β⟩}={a^θ | θ∈G_C}. This cannot hold as stated. In Definition 3.2, β acts as a cycle (Cγ_{00} Cγ_{01} … Cγ_{0n′}) with γ_{00}=1, so β maps the chosen clopen set C to the disjoint set Cγ_{01}. Since a is a nontrivial element of G_C, it has nonempty support contained in C; hence a^β has support contained in Cγ_{01}, which is disjoint from C. Thus a^β ∉ G_C. Therefore S, as defined with θ ranging over ⟨α,β⟩, contains an element not in G_C, and cannot equal {a^θ | θ∈G_C}. The subsequent claim that S is a normal subset of the simple group G_C, and that ⟨S⟩=G_C, therefore has no basis as written. This is not a mere cosmetic issue: (3.5) is the step that proves G_C≤⟨α,β⟩, which is then used to prove G=⟨α,β⟩. If the intended definition was S={a^θ | θ∈H}, where H is the setwise stabiliser of C, then the proof likely goes through, but the text as written is internally inconsistent. The authors should clarify this point; as it stands, the main theorem is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":27003,"tokens_out":8807,"duration_ms":72131,"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":[{"comment":"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.","section":"§3.1, Proposition 3.3(vi), around Eq. (3.5)"},{"comment":"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).","section":"§3.2, Proposition 3.4"}],"minor_comments":[{"comment":"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.","section":"§2.3, Proposition 2.8"},{"comment":"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.","section":"Example 3 and Corollary 3.7"},{"comment":"There is a typo: 'let x1,...,xN be the of words' should read 'let x1,...,xN be the words'.","section":"Example 8"},{"comment":"The third author's affiliation contains a typo: 'University of Birminhgham' should be 'University of Birmingham'.","section":"Author affiliations"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the false equality in Proposition 3.3(vi). It is not a matter of a missing detail but an actual mathematical error in the text. That said, the intended argument is clear enough that a repair appears straightforward, and the rest of the paper is careful and constructive. I recommend major revision rather than rejection, provided the authors fix the proof and check that Proposition 3.4 and all dependents go through with the corrected definition. I also note that the proof of Proposition 2.8 is imported from [4], which includes two of the current authors; this is not a concern given that [4] is published, but the provenance should be explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth taking seriously: Theorem 1 answers Sapir's (2,3)-generation question for Thompson's group V and gives uniform (m,n)-generation for all finitely generated simple vigorous groups, and the surrounding results (three involutions, minimal generating sets of every finite size, uniform spread at least 1, 2-generated direct powers) are genuinely new relative to [4]. The authors give explicit generators for V in Examples 3 and 5, and Theorem 2.9 is a useful new generation criterion. The writing is clear and the constructions are concrete.\n\nBut I have to flag a real gap in the proof of Proposition 3.3(vi), the step that shows G_C ≤ ⟨α,β⟩. The text defines S = {([σ^r,τ^r])^θ | θ∈⟨α,β⟩} and asserts this equals {([σ^r,τ^r])^θ | θ∈G_C}. That equality is false: β sends C to a disjoint clopen set Cγ_{01}, so a^β has support in Cγ_{01}, hence is not in G_C. Thus S as written is not a subset of G_C, and the normality/simplicity argument collapses. The fix is almost certainly to define S using H, the setwise stabiliser of C in ⟨α,β⟩; then (3.2) gives exactly the equality with G_C-conjugates, and the rest of the argument should go through. But as written, the main theorem is not fully supported at this point.\n\nTwo smaller things. Proposition 2.8, the input generation pair imported from [4, Theorem 5.15], is only outlined here; that is acceptable as a dependency on published work, but the paper should be explicit that the main theorem inherits the full proof from [4]. And the existence of a clopen C with [C]_G even is asserted without proof in Theorem 3.5 and Definition 3.2; it is probably true for vigorous groups, but should be proved or cited.\n\nOverall: if the S/H issue is repaired, I expect the theorems to stand. The paper deserves a serious referee; it is exactly the kind of work that should be sent to referees rather than desk-rejected. I would recommend asking the authors to fix Proposition 3.3(vi) and spell out the [C]_G evenness, then accept. For a reading group, it is a good paper with a nice learning opportunity about the gap.","headline":"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.","tokens_in":27565,"tokens_out":5476,"would_cite":true,"duration_ms":49067,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F05","20E32","20B27"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["simple vigorous groups","Thompson's group V","(2,3)-generated","three involutions","minimal generating sets","uniform spread","strong generation","Cantor space homeomorphisms"],"falsifier":"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.","tokens_in":1232,"feed_emoji":"♾️","tokens_out":2159,"duration_ms":70955,"temperature":0.7,"pith_summary":"This paper tries to show that the strong generation properties known for finite simple groups also hold for finitely generated simple vigorous groups—a broad class of homeomorphism groups of Cantor space that includes Thompson's group V and its relatives. Its central result is that any such group is (m,n)-generated for any m≥2 and n≥3; taking (m,n)=(2,3) gives Thompson's group V a two-element generating pair with orders 2 and 3, settling a question that had been open. Along the way the authors prove that these groups are generated by three involutions, have minimal generating sets of every size at least two, have uniform spread at least 1, and have 2-generated direct powers. Because the proofs are constructive, the generation results also yield explicit generators for V. If true, the results transfer a body of finite-simple-group generation theory into a very different infinite setting.","feed_headline":"Thompson's group V is (2,3)-generated","feed_subtitle":"A new constructive proof shows every finitely generated simple vigorous group is generated by two torsion elements, with explicit generators","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Every f.g. simple vigorous group is (2,3)-generated","Thompson's V answers Sapir's question: (2,3)-generated","Three involutions generate any f.g. simple vigorous group","Every nontrivial element lies in a generating pair","Minimal generating sets of every size for simple vigorous groups"],"cache_read_input_tokens":28672,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Every f.g. simple vigorous group is (2,3)-generated","Thompson's V answers Sapir's question: (2,3)-generated","Three involutions generate any f.g. simple vigorous group","Every nontrivial element lies in a generating pair","Minimal generating sets of every size for simple vigorous groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001651,"raw_usage":{"total_tokens":6454,"prompt_tokens":867,"completion_tokens":5587,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":611,"completion_tokens_details":{"reasoning_tokens":5512}},"tokens_in":611,"tokens_out":5587,"duration_ms":36294,"temperature":1.0,"reasoning_tokens":5512,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T06:46:29.988999+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}