{"id":"91609092-f7e4-4304-945f-1549f8e9245f","arxiv_id":"2608.05362","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Kazhdan group with a sofic embedding having an ergodic centralizer is LEF; finitely presented examples are residually finite.","lead":"The paper proves that a Kazhdan group with a sofic embedding whose centralizer acts ergodically must be locally embeddable into finite groups, and every finitely presented such group is residually finite. The main tool is a proof that such a centralizer is itself a metric ultraproduct of permutation groups.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5's uniform Becker–Chapman stability variant is load-bearing: it must hold uniformly over all finite groups F_n, yet the paper neither proves it nor cites a precise theorem in [3].","rationale":"The paper is careful and the internal argument is largely coherent; the majority arguments in Lemma 2.6, Lemma 4.4, and Proposition 4.5 are plausible, and I found no obvious internal inconsistency. The strongest claims depend on two external inputs: Kun's expander decomposition (Theorem 2.5) and the uniform Becker–Chapman stability statement. Between these, the Becker–Chapman variant is more load-bearing because Proposition 5.1 applies it to the full groups F_n, which are not a fixed finite group and can have unbounded size; the paper does not provide a proof or a precise reference. The reader's weakest_assumption identified the same point. I therefore agree with the conditional posture: the argument should not be accepted as complete until the uniform stability statement is verified in [3] or proven for the relevant class. No additional internal error was found, so the overall verdict need not change.","tokens_in":16474,"tokens_out":10705,"duration_ms":97224,"concrete_test":"Locate the exact statement in Becker–Chapman [3] and check whether it gives the uniform variant used in Section 5, with δ depending only on ε and applicable to all finite groups simultaneously. If it does not, attempt to prove the variant for the special class F_n=[[C_n]] using Remark 4.7 (wreath products of isotropy groups with symmetric groups); if no proof is found, construct a sequence of finite groups Γ_m with defect tending to 0 whose nearest exact actions on sets of size ≤(1+ε)|V_m| have distance bounded below by a positive constant, which would refute the variant and invalidate Proposition 5.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is the black-box stability statement in Section 5. Proposition 5.1 needs, for every ε>0, a δ>0 that works for every finite group Γ and every δ-approximate action f:Γ→Sym(V), producing a genuine action on a superset V'⊇V with |V'\\V|≤ε|V'| and sup_a d_H(fbar(a), f(a))≤ε. This uniformity over Γ is essential because the groups F_n=[[C_n]] are a varying family of finite full groups with |F_n|→∞; a stability theorem with δ depending on |Γ| would give no control when applying the theorem along an ultrafilter to the sequence F_n. If [3] only proves flexible stability for each fixed finite group, or only for amenable groups, then the exact equality C_{Sym(Y_n)}(π'(G))=Q_U ρ_n(F_n) in Proposition 5.1 is not established. That equality is precisely what converts the cluster-groupoid construction into Theorem 3.1's description of the centralizer as a metric ultraproduct, and Theorem A depends on it. The paper neither proves the uniform variant nor cites a theorem number, so the central claim is conditional on an unverified external result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem A: if a Kazhdan group G admits a sofic embedding into a metric ultraproduct of symmetric groups whose centralizer acts ergodically on the associated Loeb probability space, then G is locally embeddable into finite groups (LEF), and if G is finitely presented then G is residually finite. The main technical result, Theorem 3.1, asserts that after an essentially equivalent modification of the finite models, the centralizer of such a sofic embedding is a metric ultraproduct of finite permutation groups. The proof proceeds through Kun's expander decomposition, a two-component repair argument for approximate intertwiners, a cluster-groupoid construction, and Becker–Chapman flexible stability. The paper also formulates two open problems in Section 6.","tokens_in":16681,"tokens_out":15354,"duration_ms":142497,"significance":"If the proof can be made fully self-contained, the result is significant: it gives a structural obstruction to ergodic centralizers of sofic approximations of Kazhdan groups and connects the Hayes–Kunnawalkam Elayavalli conjecture with residual finiteness. The internal developments in Sections 3 and 4 are carefully structured, and the reduction in Section 2 from rigid centralizers to LEF is elegant and clearly explained. The paper is also honest about its reliance on external results, but this honesty exposes the main risk: the crucial uniform stability input in Section 5 is only asserted, not proved or precisely cited.","major_comments":[{"comment":"The stated 'uniform variant' of Becker–Chapman flexible stability is load-bearing: Proposition 5.1 applies the stability theorem to the sequence of finite groups F_n = [[C_n]] with |F_n| tending to infinity, so the same delta must work uniformly for all finite groups. If the stability modulus in [3] depends on |Gamma|, the proof cannot pass to the ultrafilter, and the equality C(pi'(G)) = prod_U rho_n(F_n) in Proposition 5.1, hence Theorem 3.1 and Theorem A, would not follow. The manuscript neither proves the uniform statement nor cites a theorem number in [3]; please supply a proof or an exact reference, including the precise hypotheses on Gamma, V, and the enlargement bound.","section":"Section 5, preamble to Proposition 5.1"},{"comment":"Proposition 3.3 is the technical engine of the cluster groupoid construction, but its proof invokes '[12, Proposition 3.3]' without restating that result. Since [12] is a preprint and the cited proposition is described only in prose, the manuscript should state it precisely, or prove it, so that the constants C_1 and rho and the hypotheses on the regular S-labelled graphs are verifiable.","section":"Section 3, Proposition 3.3"},{"comment":"Theorem 2.5, Kun's expander decomposition, is also load-bearing and is cited to the unpublished preprint [11]. The manuscript should either include a proof, state the theorem in full with its hypotheses, or cite a published version; otherwise the expander-component reduction used at the start of Theorem 3.1 rests on an external statement whose status the reader cannot verify from the present paper.","section":"Section 2, Theorem 2.5"}],"minor_comments":[{"comment":"The modification of sigma_n on (Y_n \\ X_n) cup sigma_n^{-1}(Y_n \\ X_n) to make it preserve X_n is only sketched; since the argument is needed in order to apply Proposition 4.5(c), please spell out the construction of the modified permutation and verify that its commutation defects with alpha_n(s) still tend to zero along U.","section":"Section 5, Proposition 5.1, converse direction"},{"comment":"The footnote about the OpenAI announcement is extraneous to the mathematical content of the paper and should be removed or moved to a clearly separated editorial note.","section":"Section 1, first footnote"},{"comment":"The notation pi: G -> U(prod_U M_{d_n}(C)) is terse; identifying the target as the unitary group of the tracial ultraproduct, or writing the map coordinatewise, would improve readability.","section":"Section 6, Open problem 6.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is heavily built on the authors' own preprints [11] and [12]. Please verify their publication or acceptance status before a final decision. The uniform Becker–Chapman stability statement in Section 5 is the main risk; if the authors can supply a precise citation or a proof, I would be willing to recommend acceptance after the remaining local issues are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nRead Alekseev–Thom, arXiv:2608.05362. The punchline: Theorem A is a genuinely new result — if the stated uniform Becker–Chapman stability variant holds, then the proof is convincing and the conclusions are significant. The main technical theorem, describing the centralizer of such a sofic embedding as a metric ultraproduct of permutation groups, is a substantial extension of Kun–Thom.\n\nWhat the paper does well: the cluster groupoid construction in Sections 3–4 is careful and detailed. Lemma 2.6 and Proposition 2.7 give a clean majority argument for centralizers of transitive actions. Lemma 4.2's verification that the cluster relation is a groupoid is thorough, and the distance arguments check out. I found no internal gap. The paper is honest about relying on Kun's expander decomposition and the Kun–Thom repair argument, and those are cited to specific prior results.\n\nThe soft spot is exactly where your stress-test note points: the uniform variant of Becker–Chapman flexible stability stated in Section 5 is load-bearing. Proposition 5.1 needs a δ that works uniformly over the varying finite groups F_n = [[C_n]] as n goes along the ultrafilter. The paper states this uniform variant and cites [3] without a theorem number, and does not prove it. If [3] only gives stability with δ depending on the group, or only for amenable groups, then the equality C(π'(G)) = ∏_U ρ_n(F_n) in Proposition 5.1 is not established, and with it Theorem 3.1 and Theorem A collapse. This is not a manufactured concern — the uniform statement is written down as an assumption, and the subsequent application along an ultrafilter genuinely needs uniformity. The authors should either prove that statement from [3] or give a precise reference to where it is proved.\n\nMinor: the footnote about the OpenAI announcement is irrelevant to the mathematics and will date the paper, but it does not affect the content. The self-citation pattern (Kun, Kun–Thom) is heavy but not circular; the cited results are specific and used as a technical engine.\n\nBottom line: this is a serious paper by knowledgeable authors and it deserves a serious referee. If the Section 5 dependency is resolved, it will be a valuable contribution to the theory of sofic approximations and property (T). I would not yet cite Theorem A as established, but I would send it to review.","headline":"Theorem A is new and the proof is carefully built, but the argument depends on a uniform Becker–Chapman stability variant that the paper states without proof or precise citation.","tokens_in":732,"tokens_out":1854,"would_cite":false,"duration_ms":35908,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F69","20F05","37A15","46L10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a Kazhdan group with a sofic embedding whose centralizer acts ergodically on the Loeb space must be locally embeddable into finite groups, and residually finite if finitely presented.","keywords":["sofic groups","Kazhdan property (T)","centralizers","metric ultraproducts","Loeb spaces","LEF groups","residual finiteness","stability of approximate actions"],"falsifier":"Find a sequence of approximate actions of finite groups on finite sets whose defect tends to zero but where every genuine homomorphism on a slightly enlarged set remains Hamming distance at least a fixed c>0 away; that would refute the uniform stability statement used in Section 5 and break Proposition 5.1, the step that makes Theorem 3.1 and Theorem A depend on an unproved premise.","tokens_in":16237,"feed_emoji":"🧩","tokens_out":8299,"duration_ms":69512,"temperature":0.7,"pith_summary":"The paper proves a conditional rigidity statement about sofic approximations of groups with Kazhdan's property (T). If a Kazhdan group G admits a sofic embedding into a universal sofic group whose centralizer acts ergodically on the associated Loeb probability space, then G must be locally embeddable into finite groups (LEF); when G is finitely presented, this forces G to be residually finite. The point is that the additional ergodicity demand on the centralizer, which a 2024 conjecture predicts every sofic group should satisfy, is extremely restrictive for Kazhdan groups. Its technical engine is a description of such a centralizer as essentially a metric ultraproduct of finite permutation groups (Theorem 3.1).","feed_headline":"Kazhdan groups with ergodic sofic centralizers are LEF","feed_subtitle":"Finitely presented cases must then be residually finite, narrowing the search for non-sofic groups.","key_machinery":"The central object is the cluster groupoid $\\mathcal{C}_n$ built from partial bijections between the high-expansion components of a sofic approximation. Its arrows are clusters of almost-equivariant partial bijections that are close in Hamming distance; composition is defined only after repairing the raw composition via a two-component repair lemma. The finite full group $[[\\mathcal{C}_n]]$ of total bisections of this groupoid is shown, by a two-sided majority argument, to exhaust the centralizer up to asymptotic error, and a stability theorem for near-actions of finite groups then corrects the approximate action to a genuine homomorphism $\\rho_n: [[\\mathcal{C}_n]] \\to \\mathrm{Sym}(Y_n)$. The ultraproduct of the groups $\\rho_n([[\\mathcal{C}_n]])$ is exactly the centralizer, yielding the metric-ultraproduct-of-permutation-groups description.","core_discovery":"The central claim is the paper's Theorem A: a Kazhdan group that has a sofic embedding into a universal sofic group with the property that the centralizer acts ergodically on the Loeb probability space is LEF, and if finitely presented, residually finite. The proof establishes the stronger structural statement Theorem 3.1: after replacing the finite models by essentially equivalent ones, the centralizer of the sofic embedding is exactly a metric ultraproduct of finite permutation groups $A_n \\leq \\mathrm{Sym}(Y_n)$. This rigidity of the centralizer, combined with a discreteness argument for right-translation actions of normalizer quotients, converts the embedding into a genuine embedding into an algebraic ultraproduct of finite groups, which is precisely an LEF certificate.","pith_inferences":["The theorem suggests that the ergodicity of the centralizer is a 'finite-nearness' condition: it forces the sofic model to admit a co-large subsystem on which the model is genuinely finite. One could test whether weaker invariants, such as the non-existence of invariant Loeb-measurable subsets with intermediate measure, already imply LEF.","If the recently announced proof of a non-sofic finitely presented Kazhdan group is correct, it does not contradict Theorem A; that group would simply fail the hypothesis of admitting any sofic embedding at all. It would instead shift the question to whether the ergodic-centralizer conjecture is refuted for non-sofic groups.","The two-component repair and cluster-groupoid construction is plausibly adaptable to tracial ultraproducts of matrix algebras, an extension the paper itself poses as an open problem; the permutation-group centralizer rigidity proven here is a natural template for a von Neumann algebraic centralizer rigidity statement."],"forward_implications":["Every finitely presented Kazhdan group that admits a sofic embedding with ergodic centralizer is residually finite; hence no such group can serve as a finitely presented, non-residually finite counterexample to the soficity conjecture.","Combining Theorem A with the conjecture that every sofic group admits an embedding with ergodic centralizer would make every sofic Kazhdan group LEF and every finitely presented sofic Kazhdan group residually finite, sharply constraining the geometry of sofic approximations of Kazhdan groups.","The centralizer of any sofic embedding of a Kazhdan group is, up to essentially equivalent finite models, a metric ultraproduct of finite permutation groups; this structural rigidity holds for all sofic approximations of Kazhdan groups, not just those with ergodic centralizers.","Known Kazhdan groups that are LEF but not residually finite must fail the ergodic-centralizer condition in every sofic embedding they admit."],"supporting_citations":[{"why":"Supplies the expander decomposition theorem that splits a sofic approximation of a Kazhdan group into high-Cheeger components, on which all subsequent analysis is built.","marker":"[11]"},{"why":"Provides the two-component repair lemma and the majority/boundary arguments from which the cluster groupoid construction is adapted.","marker":"[12]"},{"why":"Yields the stability theorem whose uniform variant is used to correct approximate actions of the finite full groups to genuine homomorphisms in Proposition 5.1.","marker":"[3]"},{"why":"Establishes the metric ultraproduct model of universal sofic groups, the ambient setting for the embeddings studied in the paper.","marker":"[5]"},{"why":"Formulates the conjecture that every sofic group admits an embedding with ergodic centralizer, whose interaction with Theorem A is drawn out in Corollary 5.2.","marker":"[8]"}],"fun_headline_variants":["Ergodic centralizer forces Kazhdan groups into LEF","Sofic Kazhdan with ergodic centralizer is LEF","Kazhdan groups get LEF from ergodic sofic centralizer","When Kazhdan centralizer acts ergodically, sofic implies LEF","Centralizer ergodicity turns sofic Kazhdan into LEF"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof uses a uniform stability theorem: for every epsilon there is a delta such that any map from a finite group to permutations with defect below delta is epsilon-close to a genuine homomorphism after adding a small number of new points; if this uniform statement is not a proved consequence of the cited stability results, the centralizer equality that the whole argument rests on does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Ergodic centralizer forces Kazhdan groups into LEF","Sofic Kazhdan with ergodic centralizer is LEF","Kazhdan groups get LEF from ergodic sofic centralizer","When Kazhdan centralizer acts ergodically, sofic implies LEF","Centralizer ergodicity turns sofic Kazhdan into LEF"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000428,"raw_usage":{"total_tokens":2100,"prompt_tokens":769,"completion_tokens":1331,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":385,"completion_tokens_details":{"reasoning_tokens":1233}},"tokens_in":385,"tokens_out":1331,"duration_ms":8764,"temperature":1.0,"reasoning_tokens":1233,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T14:35:52.946573+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a sequence of approximate actions of finite groups on finite sets whose defect tends to zero but where every genuine homomorphism on a slightly enlarged set remains Hamming distance at least a fixed c>0 away; that would refute the uniform stability statement used in Section 5 and break Proposition 5.1, the step that makes Theorem 3.1 and Theorem A depend on an unproved premise.","supporting_citations":[{"cited_title":"Becker and M","cited_arxiv_id":null,"evidence_quote":"Yields the stability theorem whose uniform variant is used to correct approximate actions of the finite full groups to genuine homomorphisms in Proposition 5.1."},{"cited_title":"Elek and E","cited_arxiv_id":null,"evidence_quote":"Establishes the metric ultraproduct model of universal sofic groups, the ambient setting for the embeddings studied in the paper."},{"cited_title":"Hayes and S","cited_arxiv_id":null,"evidence_quote":"Formulates the conjecture that every sofic group admits an embedding with ergodic centralizer, whose interaction with Theorem A is drawn out in Corollary 5.2."}],"review_version":1}