{"id":"ac7047fe-845c-449c-a03b-b941a5f20fdd","arxiv_id":"1909.00282","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Product groups Σ×Λ are not very flexibly P-stable when Σ admits a non-abelian free quotient and Λ lacks property (τ), so P-stability is not closed under direct products.","lead":"This paper proves that many direct products of finitely generated groups, such as F2 x Z and F2 x F2, are not stable with respect to permutations, even though each individual factor is stable. It answers an open question about whether permutation stability is preserved under direct products.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem A's proof is coherent and the Kassabov-based generalization is sound.","rationale":"The reader's ACCEPT is well supported. I independently traced the central argument: Theorem 7.1 combines the rigidity result Theorem 5.1 with the construction Lemma 6.1, and both are internally consistent. The only candidate soft point I found is not the external Kassabov dependence highlighted by the reader, but a small left/right multiplication switch: Lemma 2.6 is stated and proved for left multiplication, whereas Lemma 6.1 needs right-almost-invariant sets. This is easily fixed by inverting the constructed sets, and the fix does not disturb the density interval 1/7 to 1/6 or condition (c). Since the claimed theorem and its proof survive that repair, and since Kassabov's theorem is a published, standard expander result, I see no load-bearing concern meriting a change. The abstract's weaker formulation than Theorem A is only a presentational issue and does not affect the correctness of the main claim.","tokens_in":27385,"tokens_out":37968,"duration_ms":349715,"concrete_test":"Check the one implicit switch in the proof: apply Lemma 2.6 to the opposite group (or replace the left-invariant set C_n by C_n^{-1}) and verify that (6.1) holds with right multiplication and that condition (c) in Lemma 6.1 remains true; if either fails, the construction in Lemma 6.1 needs repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. I traced both main inputs to Theorem A. Theorem 5.1's rigidity argument legitimately applies Theorem 4.1: ε_n→0, κ=inf_n κ(X_n,p_n(S))>0, and (5.1) follows from d_H(σ_n(g),τ_n(g)|X_n)→0 together with commutation of τ_n(Γ) and τ_n(Λ); (5.2) then upgrades K_n to all of τ_n(Λ). Lemma 6.1's construction is also sound: condition (c) is a direct double-count, and the averaging over q_n(Λ) gives the required 1/126 lower bound. The only implicit point is that Lemma 2.6 produces left-almost-invariant sets while Lemma 6.1 uses right-almost-invariance; this is repaired by replacing C_n with C_n^{-1} (equivalently, applying Lemma 2.6 to the opposite group), with no effect on the density bound or condition (c). Kassabov's theorem is a published expander result, not a hidden assumption, and the proof otherwise uses only standard Selberg/Bourgain-Varju facts in the specialized cases. The abstract's weaker hypothesis relative to Theorem A is a presentational discrepancy, not a mathematical flaw.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies permutation stability (P-stability) and its flexible variants for countable groups. The main result, Theorem A, states that if a finitely generated group Σ admits a non-abelian free quotient and a finitely generated group Λ does not have property (τ), then Σ×Λ is not very flexibly P-stable. The proof combines two ingredients. First, Theorem 4.1 gives a structural rigidity statement for permutations of a finite group G that almost commute with the left regular representation; Theorem 5.1 converts this into a rigidity statement for asymptotic homomorphisms of (Γ∗Z)×Λ. Second, Lemma 6.1 constructs, from a failure of property (τ) along a sequence of finite quotients of Λ, an asymptotic homomorphism of (Γ∗Z)×Λ that is far from commuting on the Z-factor. These pieces are assembled in Theorem 7.1, and Theorem A follows by applying Kassabov's expander construction for symmetric groups in the general case and the Selberg property in the special case where Λ has an infinite cyclic quotient. The paper derives Corollary B, including non-P-stability of F_m×Z^d and F_m×F_n, thereby answering a question of Becker, Lubotzky and Thom, and Theorem D, giving examples that are not weakly very flexibly P-stable.","tokens_in":27606,"tokens_out":12570,"duration_ms":125135,"significance":"If the result stands, it is a substantial contribution to the theory of permutation stability: it settles the direct-product question of Becker, Lubotzky and Thom in the negative and provides the first non-amenable residually finite groups that are not flexibly P-stable. The paper also gives a clean separation between P-stability and Hilbert-Schmidt stability, since F_m×Z^d is HS-stable by prior work but not P-stable. The proof strategy is modular and convincing: explicit constants are tracked through Theorem 4.1 and Lemma 6.1, the rigidity argument in Theorem 5.1 is well separated from the construction in Lemma 6.1, and the key external inputs (Kassabov's expanders, Selberg property, Bourgain-Varjú expansion) are used in a transparent way. The self-contained proofs of Lemmas 2.6, 4.2 and 5.2 add to the reliability of the paper. I found no circularity or hidden ad hoc assumptions: the central claims follow from previously established results that do not include the target theorems.","major_comments":[],"minor_comments":[{"comment":"The proof invokes Lemma 2.6 to obtain sets C_n satisfying (6.1), but Lemma 2.6 produces left-almost-invariant sets, whereas (6.1) requires right-almost-invariance. This is a genuine but local gap: it is repaired by replacing C_n with C_n^{-1}, which preserves the density bounds and condition (c). I ask the author to add this one-sentence justification.","section":"§6, Lemma 6.1"},{"comment":"In the restatement of condition (1) after applying Lemma 6.1, the variables are mis-stated: it should read \"for all g∈Γ, h∈Λ, x∈X_n\" rather than \"for all g∈Σ, h∈Λ, x∈X_n\", since σ_n is defined on (Γ∗Z)×Λ and p_n is defined on Γ. The subsequent use of the condition makes the intended meaning clear, but the displayed line is confusing as written.","section":"§7, proof of Theorem 7.1"},{"comment":"The abstract states the main class as \"Σ admits a non-abelian free quotient and Λ admits an infinite cyclic quotient\", while Theorem A assumes only that Λ does not have property (τ), which is stronger. The abstract therefore understates the scope of the theorem; aligning the abstract with Theorem A would avoid confusing readers about the role of property (τ).","section":"Abstract and Theorem A"},{"comment":"There is a typo in \"flexbily P-stable\" in the proof of part (3); it should be \"flexibly P-stable\".","section":"§7, proof of Corollary B"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about this paper: it answers a named open question from Becker–Lubotzky–Thom by showing that P-stability is not closed under direct products, and it gives the first non-amenable residually finite groups that are not flexibly P-stable. That is a real advance, not a repackaging. The main theorem is broad: if Σ has a free non-abelian quotient and Λ lacks property (τ), then Σ×Λ is not very flexibly P-stable. The corollaries (free times abelian, free times free, Baumslag–Solitar in the remaining case, braid groups) follow cleanly.\n\nWhat the paper does well is the architecture. The proof separates into two reusable pieces: a rigidity theorem (Theorem 5.1) saying that an asymptotic homomorphism close to a homomorphism on a property-(τ) factor must be close to one extending the prescribed action, and a construction lemma (Lemma 6.1) producing asymptotic homomorphisms with a nontrivial obstruction from non-τ almost-invariant sets. The constants are explicit, the lemmas are stated sharply, and the reliance on Kassabov's expander family is explicit and published. The citation pattern is appropriate: the author cites his own earlier work only as a technique, not as a load-bearing result. I traced the main dependencies and found no circularity.\n\nSoft spots are minor. The abstract states a weaker hypothesis than Theorem A—it mentions infinite cyclic quotients, while the theorem only needs no property (τ)—which might confuse a reader but does not affect correctness. There is a small handedness issue in Lemma 6.1: Lemma 2.6 yields left-almost-invariant sets while the construction uses right-almost-invariance. The stress-test note is right that replacing C_n with C_n^{-1} fixes it, so this is cosmetic. The full generality does rest on Kassabov's theorem, but that is a standard published input, not a hidden assumption. The author also honestly notes that the amenable–amenable direct product question remains open.\n\nThe intended reader is a geometric group theorist or someone working on stability, sofic groups, or property (τ). This is exactly the kind of paper a serious journal should send to referees. My recommendation: engage with it, referee it, and accept after minor presentation fixes.","headline":"A genuine and important result—P-stability is not closed under direct products—proved with clean, modular arguments and honest limitations; it deserves serious refereeing and likely acceptance after minor revision.","tokens_in":28146,"tokens_out":1058,"would_cite":true,"duration_ms":27210,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20B30","22D55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that Σ×Λ is not very flexibly P-stable whenever Σ has a non-abelian free quotient and Λ lacks property (τ).","keywords":["P-stability","permutation stability","property (tau)","direct products","asymptotic homomorphisms","symmetric groups","expander graphs","flexible stability"],"falsifier":"Compute, for the asymptotic homomorphism σ_n constructed in Lemma 6.1 for F_3 × Z, the quantity max_{h ∈ Z} d_H(σ_n(t,e)σ_n(e,h), σ_n(e,h)σ_n(t,e)); the proof shows it is at least 1/126 infinitely often. If one could nevertheless find homomorphisms τ_n on larger sets with d_H(σ_n(g), τ_n(g)|_{X_n}) → 0 for every generator g, the rigidity theorem would fail and Theorem A would be open.","tokens_in":27172,"feed_emoji":"🔀","tokens_out":9353,"duration_ms":80468,"temperature":0.7,"pith_summary":"The paper establishes that permutation stability, the property that approximate homomorphisms to finite symmetric groups are always close to genuine homomorphisms, is not preserved by direct products. Its main theorem says that if Σ is a finitely generated group with a non-abelian free quotient and Λ is a finitely generated group without property (τ), then Σ × Λ is not very flexibly P-stable. In particular, since free groups and abelian groups are P-stable, the groups F_m × Z^d and F_m × F_n are not P-stable, answering an open question about direct products in the negative. The same mechanism gives the first non-amenable, residually finite groups that fail flexible P-stability, and it also settles the remaining Baumslag-Solitar cases and covers braid groups.","feed_headline":"Permutation stability is not closed under direct products","feed_subtitle":"Product of two P-stable groups can be non-P-stable, giving the first non-amenable residually finite examples.","key_machinery":"The load-bearing objects are two. The first is the rigidity theorem for asymptotic homomorphisms of (Γ ⋆ Z) × Λ into Sym(X_n), where X_n = Γ/Γ_n and Γ has property (τ) with respect to {Γ_n}: if the asymptotic homomorphism is close on X_n to the restriction of a homomorphism on larger sets, then it is close to a homomorphism that extends the original action. The second is the construction of a twist permutation ρ_n from an almost-invariant subset of X_n of density between 1/7 and 1/6, which exists whenever Λ lacks property (τ) with respect to quotients given by homomorphisms q_n : Λ → X_n. The expander result of [Ka05], that symmetric groups carry bounded-degree Cayley graphs forming an expander family, supplies the property-(τ) quotients that make the general case go through, while the special case of infinite cyclic quotients uses the Selberg property for subgroups of SL_2(Z).","core_discovery":"On the paper's own terms, the central discovery is Theorem A: for finitely generated groups Σ and Λ, if Σ admits a non-abelian free quotient and Λ does not have property (τ), then Σ × Λ is not very flexibly P-stable. The proof proceeds by fabricating asymptotic homomorphisms from (F_L ⋆ Z) × Λ to finite symmetric groups that act on coset spaces X_n = Γ/Γ_n by left-right multiplication, and then twisting one generator by a permutation ρ_n built from an almost-invariant subset of X_n. A rigidity theorem shows that any homomorphism on a larger set that approximates the twisted asymptotic homomorphism would have to be close to a homomorphism extending the original action; but the twist is constructed to remain far from every such homomorphism, with commutator distance at least 1/126 infinitely often. The consequence is that P-stability is not closed under direct products, since F_m and Z^d, and F_m and F_n, are individually P-stable while their products are not.","pith_inferences":["The proof suggests that the obstruction is the combination of a large factor, one with a non-abelian free quotient, and a factor possessing almost-invariant finite quotients; whether products of two amenable P-stable groups behave differently remains open, and the paper's methods do not reach it.","The explicit construction of asymptotic homomorphisms from products to symmetric groups can probably be transplanted to other finite quotient families with expansion, such as linear groups over shrinking moduli, to produce further non-P-stable products.","A concrete testable extension is whether F_m × F_n is also Hilbert-Schmidt unstable; the paper notes this as likely and cites supporting evidence from stability in tracial von Neumann algebras.","The commuting-subgroup phenomenon of Corollary C can be read as a finitary lifting obstruction, with the twist permutation supplying the obstruction for products of large groups with non-τ factors."],"forward_implications":["If Theorem A is correct, then F_m × Z^d and F_m × F_n are not P-stable for m,n ≥ 2 and d ≥ 1, even though each factor is P-stable.","P-stability is therefore not closed under direct products, settling the open question about direct products in the negative.","The groups covered are the first non-amenable, residually finite groups known not to be flexibly P-stable.","The Baumslag-Solitar groups BS(m,n) with |m| = |n| ≥ 2 are not P-stable, completing the classification of P-stability for these groups.","The braid groups B_n and the pure braid groups PB_n are not very flexibly P-stable for every n ≥ 3."],"supporting_citations":[{"why":"supplies the bounded-degree expander Cayley graphs on Sym(n) that produce a group Γ = F_L with property (τ), the step that makes Theorem A hold in full generality.","marker":"[Ka05]"},{"why":"defines P-stability for finitely generated groups, establishes the amenable-group framework, and poses the direct-product question that Corollary B answers.","marker":"[BLT18]"},{"why":"introduces flexible and very flexible P-stability and proves property (T) groups are not P-stable; Theorem A transfers this phenomenon to products without property (T).","marker":"[BL18]"},{"why":"provides the definitional framework for permutation stability and proves abelian groups are P-stable, giving the stable factors whose products fail stability.","marker":"[AP14]"},{"why":"delivers almost-invariant subsets of finite quotients with prescribed density, used in Lemma 2.6 to build the 1/7-to-1/6 density set for the twist permutation.","marker":"[AE10]"},{"why":"supplies, through its proof, the rigidity fact that a permutation almost commuting with the regular representation is close to a right-multiplication permutation, the core of Theorem 4.1.","marker":"[Th10]"},{"why":"gives the Kazhdan-constant criterion for property (τ) and the existence of almost-invariant subsets, which Lemma 2.6 generalizes.","marker":"[LZ03]"},{"why":"provides the Selberg property for SL_2(Z), handling the special case where Λ has an infinite cyclic quotient.","marker":"[LW93]"}],"fun_headline_variants":["P-stability not closed under direct products","Products of P-stable groups can be non-P-stable","Stable groups may multiply into unstable ones","First non-amenable residually finite non-P-stable products"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the expander result of [Ka05], that symmetric groups Sym(n) admit bounded-degree Cayley graphs forming an expander family; if that failed, the proof would only reach products whose second factor has an infinite cyclic quotient.","fun_headline_variants_meta":{"raw":{"variants":["P-stability not closed under direct products","Products of P-stable groups can be non-P-stable","Stable groups may multiply into unstable ones","First non-amenable residually finite non-P-stable products"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000452,"raw_usage":{"total_tokens":2283,"prompt_tokens":964,"completion_tokens":1319,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":1255}},"tokens_in":580,"tokens_out":1319,"duration_ms":11252,"temperature":1.0,"reasoning_tokens":1255,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:57:36.101351+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for the asymptotic homomorphism σ_n constructed in Lemma 6.1 for F_3 × Z, the quantity max_{h ∈ Z} d_H(σ_n(t,e)σ_n(e,h), σ_n(e,h)σ_n(t,e)); the proof shows it is at least 1/126 infinitely often. If one could nevertheless find homomorphisms τ_n on larger sets with d_H(σ_n(g), τ_n(g)|_{X_n}) → 0 for every generator g, the rigidity theorem would fail and Theorem A would be open.","supporting_citations":[],"review_version":1}