{"id":"aaa9e0c2-6bda-4537-a78d-aa6e659be07b","arxiv_id":"2607.11775","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"There exists a torsion-free normally poly-ℤ group of Hirsch length 14 with Out(G)=1, solving Kourovka Problem 13.23 negatively.","lead":"The authors build an explicit torsion-free supersoluble group of Hirsch length 14 whose every automorphism is inner, answering a 1995 Kourovka Notebook question in the negative. The construction shows that a normal series of infinite cyclic factors does not force outer automorphisms.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim is an explicit counter-example of Hirsch length 14. All steps—Lie-algebra model, normal-form uniqueness, definition of α, order of [α] in Out(N), self-normalising property, and the final reduction for Aut(G)—are written with concrete generators and a single 6\times6 integer matrix D whose determinant is 2. The only potential soft spot flagged by the reader is already discharged inside the paper by the BCH construction and the explicit check that α4 is inner. No external data, numerical approximation, or unstated hypothesis remains. Consequently the reader's ACCEPT verdict with high confidence stands; no adjustment is warranted.","tokens_in":14234,"tokens_out":450,"duration_ms":3980,"concrete_test":"Independently recompute the six images α(Xi) and α(Zk) under the BCH product formula of Section 2 and verify that they obey the same commutator table (N1)–(N3) and that α4 equals Inn(R-1); if any relation fails, the semidirect product is undefined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (that the generator formulas for α extend to an automorphism of N) is the natural soft spot, but the manuscript already closes it: the Baker–Campbell–Hausdorff model of N is constructed first, the images of the generators under α are shown to satisfy (N1)–(N3) by direct collection, and α4 is identified with conjugation by R-1, so α is invertible. The subsequent linear-algebra checks (det D=2, s\notin DZ6, self-normalising C4 in Out(N), and the reduction that every automorphism of G is inner) are fully explicit and do not rely on hidden hypotheses. No further load-bearing gap appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs a torsion-free supersoluble (normally poly-ℤ) group G of Hirsch length 14 with Out(G)=1, giving a negative answer to Kourovka Notebook Problem 13.23. The group is realized as the semidirect product G=⟨t⟩⋉_α N, where N is a torsion-free nilpotent group of class 3 and Hirsch length 13 built from a 6-vertex graph via a rational Lie algebra (Baker–Campbell–Hausdorff) model that yields a unique normal form. An explicit automorphism α of N is defined on generators so that [α] has order exactly 4 in Out(N) and is self-normalizing; adjoining t then forces every automorphism of G to be inner. The authors also record Z(G)≅ℤ and G_ab≅ℤ⊕(ℤ/2ℤ)^6, placing the example on the boundary left open by Menegazzo–Puglisi.","tokens_in":14426,"tokens_out":826,"duration_ms":6959,"significance":"The result settles a 30-year-old Kourovka problem that had remained open precisely in the intermediate case ρ_0(G)=1. The construction is fully explicit (graph, Lie algebra, matrix D with det D=2, concrete formulas for α) and self-contained; the key linear-algebra verifications (order of [α], self-normalizer, reduction of Aut(G) to Inn(G)) are carried out by direct calculation rather than by appeal to general machinery. This supplies a concrete counter-example of modest Hirsch length and opens the natural minimal-length question posed at the end of the paper.","major_comments":[],"minor_comments":[{"comment":"Section 3, definition of α: the verification that the six generator formulas preserve relations (N1)–(N3) is asserted by “collection”; a short expanded calculation (or a reference to a computer-algebra check) would make the step fully transparent for a reader who does not wish to recompute every commutator.","section":"Section 3"},{"comment":"Lemma 4.1: the support restrictions on the columns of S are obtained by examining non-edges; listing the non-edges used (or giving a one-line matrix argument) would shorten the verification that S must be diagonal.","section":"Lemma 4.1"},{"comment":"Throughout: a few typographical slips appear (missing spaces after periods, occasional “Wegive”-style concatenations in the abstract). A light copy-edit would improve readability.","section":null},{"comment":"Question 5.3: it would be useful to record the best lower bound currently known (e.g., from the Menegazzo–Puglisi constraints) so that the gap between 14 and the theoretical minimum is explicit.","section":"Question 5.3"}],"recommendation":"accept","confidential_remarks":"The manuscript is a clean, self-contained solution of a well-known Kourovka problem. The only soft spot flagged by the reader (that α extends to an automorphism) is already closed by the BCH model and the identification α^4=Inn(R^{-1}). I see no reason to delay acceptance; the minor presentational points can be handled at the proof stage."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper closes Kourovka Problem 13.23 with a concrete torsion-free supersoluble group G of Hirsch length 14 that has a full normal series with infinite cyclic factors and Out(G)=1. That is the one fact that matters: previous work of Menegazzo–Puglisi handled ρ0≥2 and ρ0=0, and Robinson’s earlier examples were not normally poly-ℤ. The construction sits exactly on the narrow boundary they identified (Z(G)≅ℤ, Gab≅ℤ⊕(ℤ/2ℤ)6).\n\nWhat they do well is keep everything explicit. N is built from a rational Lie algebra of class 3 attached to a six-vertex graph; the BCH formula gives a unique normal form, so N is torsion-free of Hirsch length 13. The automorphism α is written on generators, shown to preserve the relations by direct collection, and proved to satisfy α4=Inn(R-1) while α2 is not inner (the vector s lies outside the image of the incidence matrix D of det 2). The self-normalising property of ⟨[α]⟩≅C4 in Out(N) is a clean linear-algebra argument over ℤ and ℚ, and the final reduction that every automorphism of the semidirect product G=⟨t⟩⋉αN is inner follows the standard strategy without gaps. The stress-test note is right: the natural soft spot (whether α really extends) is already closed by the model.\n\nSoft spots are minor and presentational. A couple of verifications (that α preserves every relation, uniqueness of the surjection G→ℤ) are left a little terse, and the Hirsch length 14 is not claimed to be minimal; the authors themselves ask what the minimal length is. None of this threatens the existence claim.\n\nThe paper is for people who work on outer automorphisms of infinite soluble or polycyclic groups. The calculations are fully displayed, so a careful reader can check them by hand. It deserves a serious referee and should be accepted after ordinary polishing. I would cite it when the topic comes up and would bring it to reading group if we are doing infinite groups this term.","headline":"Explicit counter-example of Hirsch length 14 that settles Kourovka 13.23 at the exact ρ0=1 boundary left open by Menegazzo–Puglisi.","tokens_in":15062,"tokens_out":537,"would_cite":true,"duration_ms":5330,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F28","20F16"],"pacs":[],"model":"grok-4.5","headline":"There exists a torsion-free group of Hirsch length 14 with a normal series of infinite cyclic factors and no outer automorphisms.","keywords":["outer automorphism","supersoluble group","polycyclic group","normally poly-Z","Kourovka Notebook","torsion-free nilpotent","Hirsch length"],"falsifier":"An explicit computation showing that the proposed images under α fail to preserve one of the commutator relations (N1)–(N3), or that α^{2} is in fact inner in N, or that some automorphism of the resulting semidirect product G is outer.","tokens_in":15102,"feed_emoji":"∞","tokens_out":828,"duration_ms":6438,"temperature":0.7,"pith_summary":"The paper answers a long-standing question in group theory by producing an explicit counterexample: a torsion-free group G of Hirsch length 14 that admits a finite chain of normal subgroups with every successive factor infinite cyclic, yet every automorphism of G is inner. Such groups are called normally poly-ℤ (or torsion-free supersoluble of a special kind). Earlier work had shown that any such group with abelianization rank at least 2 must have outer automorphisms, and that examples with trivial outer automorphism group can exist when the abelianization is finite; the remaining open case was precisely rank 1. The construction realises that boundary case: G is a semidirect product of a carefully chosen nilpotent group N of class 3 and Hirsch length 13 by an infinite cyclic group generated by an automorphism α of order 4 in Out(N). The resulting G has centre isomorphic to ℤ, abelianization ℤ⊕(ℤ/2ℤ)^{6}, and Out(G)=1. The existence of even one such group settles the Kourovka problem in the negative and shows that the rigidity phenomenon can occur for normally poly-ℤ groups.","feed_headline":"A length-14 torsion-free group with no outer automorphisms","feed_subtitle":"Explicit counterexample settles a 1995 Kourovka problem on normally poly-Z groups","key_machinery":"The nilpotent group N of class 3, Hirsch length 13, presented by generators X_i, Z_k, C subject to commutator relations read off a six-vertex graph, together with the explicit automorphism α that acts as -I on N_ab and satisfies α^{4} equal to conjugation by a fixed element R of N while α^{2} is not inner. The semidirect product G=⟨t⟩⋉_α N then has the desired normal series and Out(G)=1.","core_discovery":"There exists a torsion-free group G of Hirsch length 14 that admits a finite normal series with every factor infinite cyclic and yet Out(G)=1. Moreover Z(G)≅ℤ and G_ab≅ℤ⊕(ℤ/2ℤ)^{6}. The group is realised as the semidirect product ⟨t⟩⋉_α N where N is a torsion-free nilpotent group of class 3 and Hirsch length 13 constructed from a rational Lie algebra associated with a six-vertex graph, and α is an automorphism of N of order exactly 4 in Out(N).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Torsion-free supersoluble group of Hirsch length 14 with Out=1","Length-14 torsion-free normally poly-Z group with trivial Out","Hirsch-14 torsion-free group with cyclic normal series but Out=1","Torsion-free poly-Z group of length 14 and Out(G)=1","Supersoluble torsion-free G of h(G)=14 with Out(G)=1"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The map α defined on the generators of N really extends to a group automorphism of N (checked only by verifying that the images satisfy the defining relations and that its fourth power is conjugation by an element of N).","fun_headline_variants_meta":{"raw":{"variants":["Torsion-free supersoluble group of Hirsch length 14 with Out=1","Length-14 torsion-free normally poly-Z group with trivial Out","Hirsch-14 torsion-free group with cyclic normal series but Out=1","Torsion-free poly-Z group of length 14 and Out(G)=1","Supersoluble torsion-free G of h(G)=14 with Out(G)=1"]},"model":"grok-4.5","effort":"low","cost_usd":0.007398,"raw_usage":{"total_tokens":1749,"prompt_tokens":678,"num_sources_used":0,"completion_tokens":112,"cost_in_usd_ticks":73980000,"prompt_tokens_details":{"text_tokens":678,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":959,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":678,"tokens_out":112,"duration_ms":7339,"temperature":1.0,"reasoning_tokens":959,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T03:15:49.704622+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit computation showing that the proposed images under α fail to preserve one of the commutator relations (N1)–(N3), or that α^{2} is in fact inner in N, or that some automorphism of the resulting semidirect product G is outer.","supporting_citations":[],"review_version":1}