{"id":"57264872-f586-4484-a7a0-b8e7e79c0e64","arxiv_id":"2512.09180","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every countable (recursively presented) group embeds into a finitely presented, property (T), Frobenius-stable group whose second cohomology is nonzero for every unitary representation.","lead":"This paper builds many finitely presented groups with property (T) that are Frobenius stable yet fail the stronger property (T₂), and it shows (T₂) is not preserved under quotients. The construction embeds every countable group into such a group, so the examples are abundant and flexible.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A depends on Lemma 3.1, quoted without proof from the author's [FFS25]; if that lemma fails at the stated generality, the H²-nonvanishing mechanism collapses.","rationale":"The reader identified Lemma 3.1 as the weakest assumption, and I agree: it is the single step where a hidden failure would break the central theorem. Proposition B, Lemma 2.1, and the non-Frobenius-approximable input from [DCGLT20] appear coherent; the index of concern is not the overall strategy but the deferred proof of a technically delicate group-theoretic lemma. The concern is not an accusation of dishonesty—self-citation is normal—but the support is not contained in the manuscript, and the lemma's conclusion is strong: it asks for an infinite cyclic subgroup with a Cohen–Lyndon triple, whereas standard filling theorems require non-virtually-cyclic 'suitable' subgroups. Thus the transfer from [FFS25] must be checked carefully. I also note the missing post-scriptum promised in the abstract; this is a completeness issue but secondary. Since the reader's CONDITIONAL verdict already reflects exactly this dependency, no change to the verdict is needed, but the condition should be discharged either by including a proof of Lemma 3.1 or by stating the precise hypotheses under which it is imported from [FFS25].","tokens_in":8659,"tokens_out":15534,"duration_ms":164390,"concrete_test":"Obtain [FFS25, Lemma 3.5] and check whether it proves Lemma 3.1 in full generality. Additionally, run a direct verification in the exact class used in Proposition B: take Γ = Λ0 * K with Λ0 a hyperbolic property-(T) group and K any finitely generated group; pick a hyperbolic element g ∈ Λ0 whose cyclic subgroup is malnormal (e.g., no proper roots), and verify via [Sun20] that (Γ,⟨g⟩,⟨g⟩) is Cohen–Lyndon and the quotient is hyperbolic relative to the image of K with injectivity on K. If the direct check reproduces the lemma, the concern is resolved; if the cited lemma requires additional assumptions, the proof of Proposition C must be modified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem A rests on Proposition C, whose proof uses Lemma 3.1 to produce an infinite cyclic Λ such that (Γ,Λ,Λ) is a Cohen–Lyndon triple and the Dehn-filled quotient Γ/⟨⟨Λ⟩⟩ is hyperbolic relative to the image of K, injectively on K. This is the exact step that forces H²(Γ̄;V)≠0 for every unitary V: via Lemma 3.2 it injects H¹(Λ;V)≅V into H²(Γ̄;V). The proof of Lemma 3.1 is not in the paper; it is deferred to [FFS25, Lemma 3.5], the author's own preprint. The gap is not merely cosmetic: Λ is required to be infinite cyclic, whereas the 'suitable' subgroups used in the other filling arguments (Proposition 2.4, via [Osi10]/[CIOS23]) are by definition not virtually cyclic. So the lemma is doing non-obvious work—it must find a malnormal/clean cyclic subgroup whose normal closure is freely generated by conjugates and whose filling preserves relative hyperbolicity to K. If [FFS25, Lemma 3.5] contains an extra hypothesis (e.g. torsion-free or K malnormal) or an error at this point, then Proposition C's H²-nonvanishing assertion, and hence Theorem A's 'very far from T²' part, would not follow. The abstract's promised post-scriptum is also absent from the supplied text, but the mathematical dependency is the substantive issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem A: every countable (resp. recursively presented) group embeds into a (finitely presented) group Γ with property (T) that is Frobenius stable but fails property (T²) in the strongest possible way, namely H²(Γ;V)≠0 for every unitary Γ-representation V. The construction is two-stage. Proposition B uses small cancellation over relatively hyperbolic groups to produce, from a non-Frobenius-approximable group A, a group Γ that is hyperbolic relative to a subgroup K containing a prescribed countable group C, has property (T), and has no non-trivial Frobenius-approximable quotients; such groups are automatically Frobenius stable. Proposition C then uses a cyclic Dehn filling and a Cohen–Lyndon triple to produce a quotient with non-vanishing H² for all unitary coefficients. As a by-product, Corollary D shows that property [T²] does not pass to hyperbolic quotients.","tokens_in":8985,"tokens_out":10693,"duration_ms":101902,"significance":"If correct, this is a substantial contribution: it provides a large class of finitely presented property (T) groups that are Frobenius stable for the vacuous reason that they admit no non-trivial approximable quotients, while being maximally far from having property (T²). It also demonstrates that higher property (T) behaves very differently from property (T) under quotients. The exposition is clear, the small-cancellation and cohomological arguments are coherent, and the paper is careful about attributing prior work and about its open questions. The main weakness is that the decisive Lemma 3.1 is imported without proof from the author's own unpublished preprint, and the abstract promises a post-scriptum that is absent from the submitted text.","major_comments":[{"comment":"Lemma 3.1 is load-bearing for Proposition C and hence for Theorem A, but its proof is only a citation to the author's own preprint [FFS25, Lemma 3.5]. The lemma is non-obvious: it asserts the existence of an infinite cyclic Λ with (Γ,Λ,Λ) a Cohen–Lyndon triple whose Dehn filling is hyperbolic relative to the image of K and injective on K, while the standard 'suitable' subgroups in [Osi10] are not virtually cyclic. If [FFS25] is not yet available or carries extra hypotheses (e.g. torsion-free, malnormal K), Proposition C's strong non-vanishing H²(Γ̄;V)≠0 collapses. Please include a full proof, or state precisely the hypotheses and a publicly available reference with proof, or otherwise justify this step independently.","section":"Section 3, Lemma 3.1"},{"comment":"The abstract promises 'A post-scriptum relates and comments on an experience this paper had with an AI benchmark.' The submitted text contains no such post-scriptum. This is not merely cosmetic: the manuscript as submitted does not contain a section advertised by its own abstract. Please either include the promised post-scriptum or delete this sentence.","section":"Abstract / full text"},{"comment":"The finite-presentability clause 'if A has solvable word problem' relies on [BH74], which applies only to finitely generated groups. As stated, Proposition B would justify finite presentability for any A with solvable word problem, including infinitely generated ones; the proof does not cover that case. This does not affect Theorem A, since the A constructed there is a finite central extension of a finitely presented linear group, but Proposition B should be reworded to assume A is finitely generated with solvable word problem.","section":"Proposition B and Remark 2.5"}],"minor_comments":[{"comment":"Typo: 'the the family of finite-dimensional unitary groups' should be 'the family of finite-dimensional unitary groups'.","section":"Introduction, §1"},{"comment":"State explicitly that when V is viewed as a Γ-module, the action factors through Γ̄, so V is a trivial Λ-module. This is used in the proof of Proposition C.","section":"Lemma 3.2"},{"comment":"Λ is used both for a hyperbolic group in Proposition 2.3 and for a normal subgroup in Proposition 2.4; consider different letters to avoid confusion in the proof of Proposition B.","section":"Notation"},{"comment":"The sentence 'Their second Betti number vanishes ... so they also have [T²]' is confusing in light of the preceding sentence already asserting [T²]; please clarify which property is being inferred from the Betti number vanishing.","section":"Corollary D"},{"comment":"Reference [FFS25] should indicate the current version or arXiv date, since it is cited for a key lemma that is not proved in the present paper.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The mathematical skeleton appears sound and the result is potentially important for the study of stability and higher property (T). The decisive issue is external dependence: Lemma 3.1 is quoted from the author's own unpublished preprint, and it is exactly the step that produces the non-vanishing H². I recommend requesting a self-contained proof or a verifiable published reference in the revision. The missing post-scriptum promised in the abstract should also be fixed. The Proposition B finite-presentability issue does not affect Theorem A but should be corrected. I see no grounds for rejection if the lemma is supplied or verified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it cover to cover. The headline result is real and worth taking seriously: every countable (recursively presented) group embeds into a finitely presented property (T) group that is Frobenius stable but has H²(Γ;V) ≠ 0 for every unitary representation V. That answers Gonen Cohen's question in the negative and makes the class of examples much richer than the earlier virtually free and higher-rank lattice cases. The method is a deliberate combination: build groups with no non-trivial Frobenius-approximable quotients so stability is vacuous, then Dehn fill an infinite cyclic subgroup to force H² nonvanishing. Corollary D, a hyperbolic group with (T²) having a hyperbolic quotient without (T²), is a clean bonus.\n\nWhat the paper does well: it is short, explicit, and the architecture is transparent. Proposition B is a serviceable general construction using the known non-approximable group of De Chiffre–Glebsky–Lubotzky–Thom and Boone–Higman embeddings. Proposition C is the core: given a relatively hyperbolic property (T) group, Lemma 3.1 supplies an infinite cyclic Λ so that (Γ,Λ,Λ) is a Cohen–Lyndon triple; then the excision theorem gives an exact sequence H¹(Γ;V) → H¹(Λ;V) → H²(Γ̄;V), and with H¹(Γ;V)=0 and H¹(Z;V)≅V, you get nonvanishing for every unitary V. The cohomology argument is correct.\n\nThe soft spot is exactly where the reviewer expected it: Lemma 3.1 is quoted without proof from the author's own preprint [FFS25, Lemma 3.5]. That lemma does non-obvious work—it is not a standard 'suitable subgroup' statement, since suitable subgroups are by definition not virtually cyclic. If the lemma fails, or has extra hypotheses, Proposition C and hence Theorem A collapse. The paper gives no proof sketch, only a reference. That is a legitimate reason to be conditional. Also, the abstract promises a post-scriptum about an AI benchmark that is missing from the supplied text; a minor completeness issue but worth correcting.\n\nWho this is for: anyone working on stability, higher property (T), or cohomology of groups. The construction is novel and the result is significant. The citation to [FFS25] should be checked carefully by a referee, and I would strongly suggest the author include the lemma or a proof sketch in this paper. Still, the paper deserves peer review, not a desk rejection. I'd read it again and cite it once the lemma is settled.","headline":"The main construction is significant and the argument is sound, but it depends on a deferred lemma from the author's own preprint, which is the one thing to check.","tokens_in":9511,"tokens_out":7185,"would_cite":true,"duration_ms":59555,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F67","20J06"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every countable group there is a finitely presented overgroup with property (T) that is Frobenius stable yet has nonzero second cohomology against every unitary representation, and the same mechanism shows property (T^2) does not pass t","keywords":["Frobenius stability","approximable quotients","property (T)","higher property (T^2)","relatively hyperbolic groups","Cohen–Lyndon triple","Dehn filling","small cancellation"],"falsifier":"For the quotient Γ̄ produced in Proposition C, compute H^2(Γ̄; V) for a single nontrivial unitary representation V. The theorem says it is always nonzero; a single zero would refute the central claim. Alternatively, exhibit a nontrivial Frobenius-approximable quotient of a group built by Proposition B, which would destroy the claimed (vacuous) stability.","tokens_in":8499,"feed_emoji":"🧩","tokens_out":10932,"duration_ms":94915,"temperature":0.7,"pith_summary":"The paper's central result is a construction: any countable group can be embedded into a finitely presented group Γ that has property (T) and is Frobenius stable, yet is as far as possible from the higher version of property (T). Specifically, for every unitary representation V of Γ, the second cohomology H^2(Γ;V) is nonzero, whereas property (T^2) would require all these to vanish. The groups are Frobenius stable for a potentially vacuous reason: they admit no nontrivial Frobenius-approximable quotients at all. The same method proves that property (T^2) does not pass to quotients, even for hyperbolic groups.","feed_headline":"Every countable group embeds in a Frobenius-stable property (T) group","feed_subtitle":"Second cohomology stays nonzero for every unitary representation, and quotients can lose (T^2).","key_machinery":"The central objects are (1) the notion of an approximable quotient: a quotient that is the image of an asymptotic homomorphism to finite unitary groups with the Frobenius norm that stays away from the identity on nontrivial elements; a group with no such quotients is automatically Frobenius stable. (2) The Cohen–Lyndon triple (Γ, Λ, Λ), where the normal closure of an infinite cyclic subgroup Λ is a free product of conjugates; the excision isomorphism for such triples yields the exact sequence H^1(Γ;V) → H^1(Λ;V) → H^2(Γ̄;V), so property (T) (which forces H^1(Γ;V)=0) combined with Λ ≅ Z (which gives H^1(Λ;V) ≅ V) injects every unitary representation into the second cohomology of the Dehn-fill","core_discovery":"The paper establishes Theorem A: every countable (recursively presented) group embeds into a (finitely presented) group Γ that has property (T), is Frobenius stable, and does not have property (T^2); in fact H^2(Γ;V) ≠ 0 for every unitary Γ-representation V. The proof splits into two independent constructions. First, using small-cancellation quotients of relatively hyperbolic groups, the author forces a simple non-approximable subgroup to normally generate the entire group, which kills all nontrivial approximable quotients and thereby makes the group vacuously Frobenius stable while preserving property (T) and relative hyperbolicity. Second, via a Cohen–Lyndon Dehn filling along an infinite","pith_inferences":["The stability of the constructed groups is 'vacuous' in the sense that there are no nontrivial Frobenius-approximable quotients to be stable against; the real test of stability as a phenomenon lives in the residually finite setting, which the paper leaves open.","The Cohen–Lyndon filling step is a general mechanism: any group whose first cohomology vanishes into a class of modules, and which has a suitable peripheral cyclic subgroup, yields a quotient whose second cohomology is nonzero into that whole class. This could be exported to other relatively hyperbolic or acylindrically hyperbolic settings to obstruct higher cohomological vanishing.","Because property (T) itself passes to quotients while property (T^2) does not, higher cohomological vanishing is a strictly more fragile property; this suggests that any useful 'higher Kazhdan' notion may need to be formulated with quotient behaviour in mind rather than as a naive cohomological vanishing condition.","If the nonzero class in H^2 is the obstruction to defect diminishing, these groups would be Frobenius stable with a superlinear rate; the paper explicitly notes its non-vanishing theorem does not by itself show this, so the rate question is a natural next target."],"forward_implications":["Finitely presented Frobenius stable groups with property (T) exist in profusion: the construction works over every countable group, and recursively presented groups yield finitely presented examples.","Property (T^2), and its stronger variant [T_2], does not pass to quotients: a hyperbolic group with [T_2] can have a hyperbolic quotient without (T^2).","The groups produced are not residually finite — they have no nontrivial finite quotients at all — so the known stable groups remain either virtually free, higher-rank lattice-like, or non-residually finite.","The non-vanishing extends to L^1 coefficients, linking the construction to fixed-point theorems for actions on low-dimensional contractible complexes.","Question E remains open: whether a finitely presented residually finite Frobenius stable group without property (T^2) exists; the paper suggests SL(3,Z) as a candidate."],"fun_headline_variants":["Countable groups embed in Frobenius-stable (T) groups lacking (T^2)","Property (T) and Frobenius stable, but no (T^2) in new groups","Quotients destroy (T^2) for these Frobenius-stable groups","Every countable group sits in a (T) group that fails higher (T^2)"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument rides on the lemma that every non-elementary relatively hyperbolic group without nontrivial finite normal subgroups has an infinite cyclic subgroup whose normal closure is a free product of conjugates and whose Dehn-filled quotient is still relatively hyperbolic and injective on the peripheral subgroup; if that lemma fails, the strong non-vanishing of second cohomology does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Countable groups embed in Frobenius-stable (T) groups lacking (T^2)","Property (T) and Frobenius stable, but no (T^2) in new groups","Quotients destroy (T^2) for these Frobenius-stable groups","Every countable group sits in a (T) group that fails higher (T^2)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000477,"raw_usage":{"total_tokens":2125,"prompt_tokens":590,"completion_tokens":1535,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":334,"completion_tokens_details":{"reasoning_tokens":1439}},"tokens_in":334,"tokens_out":1535,"duration_ms":12176,"temperature":1.0,"reasoning_tokens":1439,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T17:28:56.592396+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the quotient Γ̄ produced in Proposition C, compute H^2(Γ̄; V) for a single nontrivial unitary representation V. The theorem says it is always nonzero; a single zero would refute the central claim. Alternatively, exhibit a nontrivial Frobenius-approximable quotient of a group built by Proposition B, which would destroy the claimed (vacuous) stability.","supporting_citations":[],"review_version":1}