{"id":"b7a09a62-6957-49c3-85a4-0bff541fd041","arxiv_id":"2608.06222","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any infranormal, nonnormal Kazhdan subgroup Gamma of a Kazhdan group G, the generalized wreath product over G/Gamma is nonsofic, with explicit residually finite examples.","lead":"Two mathematicians used the recent OpenAI proof that some groups cannot be approximated by finite permutations to build many more such groups, including explicit examples made from matrix groups over polynomial rings. Their new groups are residually finite and have Kazhdan's property (T), so they are contrarian objects in an area that usually behaves well.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem A rests on Lemma 4.2, a joint-scale cluster-groupoid lemma whose key part (4) is imported from the unpublished preprint [1] without proof; until that lemma is fully established, the main nonsoficity claim remains conditional.","rationale":"The paper's self-contained parts, including Proposition 3.1, Theorem C, Corollary D, and the explicit examples in Theorem E, appear mathematically sound. The elementary lemmas in Section 2 are correct, and the infranormality verification for the polynomial and Laurent polynomial examples is convincing. However, Theorems A and B both depend on Theorem 4.1, whose proof is built directly on Lemma 4.2. Lemma 4.2 is not proved in the text: its part (4), the representation of centralizer elements by patched total bisections of the cluster groupoid, is asserted with a one-sentence reference to the two-sided majority argument and delegated to the unpublished preprint [1]. The present text supplies only the scale bookkeeping and the distance-gap argument, not the underlying repair theorem or the representation theorem. Thus the central result is conditional on the companion preprint being correct and publicly available. I found no internal contradiction in the present argument, and I do not see a different, more load-bearing flaw: the main gap is precisely the deferred Lemma 4.2. The reader's verdict of CONDITIONAL is therefore appropriate, and my stress-test does not change it.","tokens_in":1209,"tokens_out":1504,"duration_ms":150514,"concrete_test":"Independently derive Lemma 4.2(4) from the expander-decomposition theorem [13, Theorem 1]: take an arbitrary element v = (v_n) in C_SU(sigma(Gamma)), run the two-sided majority argument componentwise on the Gamma-expander partition, and exhibit a total bisection of C_n whose patched permutation agrees with v_n on a set of density 1. Verify that all accumulated errors, including source and range defects, equivariance defects, and the completion over the global exceptional set, are o_U(1). If this derivation cannot be completed without an additional hypothesis on the Gamma-action, or if the joint-scale diagonalization in Lemma 4.2 fails for the finitely many compressors used in Theorem 4.1, then Theorem 4.1 needs a new argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 4, Lemma 4.2 states a joint-scale version of the cluster-groupoid construction, and its proof is only scale bookkeeping plus a reference to [1, Proposition 3.3, Lemma 3.4, Definition 4.1 and Proposition 4.5]. Part (4) is the load-bearing bridge: every element of C_SU(sigma(Gamma)) is represented, up to o_U(1), by patched total bisections of the finite cluster groupoid C_n, and conversely every such patched bisection centralizes sigma(Gamma). Theorem 4.1 uses this representation twice: first to represent an arbitrary centralizer element v, and then to build the conjugate permutation b_n from a partial bisection and complete it to a total bisection. Lemma 4.3 also depends on Lemma 4.2's joint scale choice and on the component matching from Proposition 3.1. If the representation theorem in [1] does not hold in this quantified joint-scale form, or if the two-sided majority argument fails for a finite family of compressors, then the cardinality/fullness argument in Theorem 4.1 collapses and Theorem A loses its foundation. This is a missing support in the present manuscript, not an internal inconsistency: the paper explicitly defers the core construction to an unpublished companion preprint. The reader's weakest-assumption analysis identifies exactly this point, and I agree.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies generalized wreath products of the form (Z/2Z) ≀_{G/Γ} G, where Γ is an infranormal, non-normal subgroup of a Kazhdan group G. Its main result, Theorem A, asserts that such wreath products are nonsofic when both Γ and G have Kazhdan's property (T). The proof strategy is to prove a permutation-centralizer normalization theorem (Theorem 4.1): for any sofic representation σ of G, the centralizer C_{S_U}(σ(Γ)) is normalized by σ(G). From this, the authors derive nonsoficity of the wreath product, a centralizer normality statement (Theorem B), and consequences for sofic p.m.p. actions, including a negative answer to Păunescu's question via generalized Bernoulli actions (Theorems C and D, Corollary D). The paper also constructs explicit residually finite Kazhdan examples over polynomial and Laurent polynomial rings (Theorem E).","tokens_in":16545,"tokens_out":9261,"duration_ms":99798,"significance":"If the main theorem is correct, it is a substantial result: it yields explicit residually finite Kazhdan groups whose generalized wreath products are nonsofic, and it answers Păunescu's question in the negative. The paper has clear strengths: Theorem E is concrete, the reduction of nonsoficity to centralizer normalization is elegant, and the scale and component-matching estimates in Sections 3 and 4 are generally coherent. The paper also gives credit to the prior expander-decomposition mechanism [13] and builds on the first nonsofic group construction [15]. However, the decisive cluster-groupoid representation theorem is not proved in the manuscript; Lemma 4.2(4) is imported from an unpublished companion preprint [1]. This makes Theorems A and B conditional on that companion. The stress-test concern lands: the load-bearing step is genuinely deferred, not merely summarized.","major_comments":[{"comment":"Part (4) of Lemma 4.2 is the load-bearing assertion that every element of C_{S_U}(σ(Γ)) can be represented, up to o_U(1) in Hamming distance, by permutations obtained from patched total bisections of the finite cluster groupoid C_n, and conversely that every such patched bisection centralizes σ(Γ). Theorem 4.1 uses this representation twice: first to represent an arbitrary centralizer element v, and then to construct the completed permutation b_n from a partial bisection. Lemma 4.3 also depends on the joint-scale choice of Lemma 4.2. The proof of Lemma 4.2, however, is only scale bookkeeping plus references to [1, Proposition 3.3, Lemma 3.4, Definition 4.1 and Proposition 4.5]; the two-sided majority argument for (4) is not carried out. As the manuscript stands, Theorems A and B are conditional on an unpublished companion. The authors should either include a complete proof of this joint-scale representation theorem or explicitly restate the main results as conditional on [1] and explain which parts of [1] are used.","section":"§4, Lemma 4.2"},{"comment":"The proof of the distance gap in part (1) is not valid as written. For two allowed partial bijections b and c with agreement set A, the asserted bound |∂_S A| ≤ 2 ε_n |Q_{n,i}| does not follow from the allowedness hypotheses: a vertex sx can lie outside A because b(sx) ≠ c(sx), and the individual equivariance defects of b and c do not control the number of such disagreement vertices. The dichotomy between d ≤ q_n and d ≥ 1 − q_n is a nontrivial structural property of the cluster groupoid and is used repeatedly: in transitivity of the equivalence relation, in well-definedness of composition, in faithfulness of F_{ℓ,n}, and in the completion step of Theorem 4.1. This point needs a complete proof or a precise reference to a proved statement in [1].","section":"§4, Lemma 4.2(1)"}],"minor_comments":[{"comment":"The phrase 'breakthrough of OpenAI' is informal and could be replaced by 'the construction of [15]' or 'the result of [15]' throughout the paper.","section":"§1, Abstract and Introduction"},{"comment":"The cluster groupoid C_n ⇒ I_n and the notions of allowed arrows, patched bisections, and total bisections are used extensively, but the paper does not restate their definitions. Since Lemma 4.2 and Lemma 4.3 rely on [1], the paper should at least summarize these notions so that a reader can follow Theorem 4.1 without consulting [1].","section":"§4, Lemma 4.2"},{"comment":"Reference [1] is listed as 'submitted to arXiv'; if it is indeed a preprint, an arXiv number and date should be given, and the paper should state explicitly which results of [1] are used and which are proved here.","section":"References"},{"comment":"In the residual-finiteness argument, the sentence 'choose m so that the finitely many exponent vectors in its support remain distinct modulo m' is correct but should be stated more explicitly: one chooses m separating all pairs of exponent vectors appearing in the chosen nonzero entry of u − I_r.","section":"§5.2, Proof of Theorem E"}],"recommendation":"major_revision","confidential_remarks":"The central concern is verifiability: Lemma 4.2(4), on which Theorems A and B rest, is deferred to an unpublished companion preprint. I would not recommend acceptance until either the full proof is included or the manuscript is explicitly framed as conditional on [1]. The reliance on the OpenAI preprint [15] is also worth editorial attention, though it does not by itself affect the internal logic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a real contribution to the nonsoficity program, with new theorems and explicit examples, but the main theorem is conditional on an unpublished companion preprint [1]. The paper's own Lemma 4.2, which carries the load-bearing cluster-groupoid representation, is not proved here; its part (4) is imported from [1]. Until that preprint is public and verified, Theorem A should be treated as conditional, not established.\n\nWhat's new: the notion of infranormal subgroup is useful; Theorem A extends the OpenAI rigidity mechanism to generalized wreath products over explicit residually finite Kazhdan groups; Theorem C and Corollary D give the promised negative answer to Paunescu's question. The examples in Theorem E are concrete and the verification is careful — the residue-field reduction argument for residual finiteness is clean.\n\nThe paper does a lot well. The proof of Proposition 3.1 is actually in the text, and the estimates in Section 4 are coherent. The authors are transparent about the dependency on [1], which is more than many papers do. The writing is dense but the structure is clear.\n\nSoft spots, in proportion: the big one is Lemma 4.2. The proof given is scale bookkeeping; the repair theorem and the representation of centralizers by patched bisections live in [1]. Lemma 4.3, Theorem 4.1, and hence Theorem A all rest on that. This is missing support in the manuscript, not an internal contradiction. A second, smaller issue is the citation load: much of the machinery comes from the same authors' earlier work [13], [14], [15], [1]. That is not by itself a flaw, but it means a referee needs to read those papers carefully to see what is genuinely new here. A minor point: the acknowledgements mention an interactive proof assistant, but no formal artifacts are provided, so that line is just disclosure.\n\nWho is this for? Researchers working on soficity, property (T), and group actions. The reader who will get value is one who wants to know where the OpenAI construction can be pushed. It deserves a serious referee — conditional acceptance at most, with the companion preprint as the condition. I would not desk-reject.","headline":"Serious and likely important, but the main theorem is conditional on an unpublished companion preprint: Lemma 4.2's core representation is imported from [1], not proved here.","tokens_in":17158,"tokens_out":1700,"would_cite":true,"duration_ms":16544,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F69","22D55","37A15","37A20","46L10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A generalized wreath product over a coset space is nonsofic when the subgroup is infranormal, non-normal, and both groups have Kazhdan's property (T), and explicit residually finite pairs realize these hypotheses.","keywords":["sofic groups","Kazhdan's property (T)","infranormal subgroups","compression semigroup","generalized wreath products","sofic actions","expander decompositions","residually finite groups"],"falsifier":"Exhibit a sofic approximation of one of the explicit groups in Theorem E in which the centralizer of $\\sigma(\\Gamma)$ fails to be normalized by $\\sigma(G)$, or find a sofic p.m.p. action of that group whose $\\Gamma$-fixed algebra is not $G$-invariant; either example would contradict Theorem C and therefore break the chain leading to Theorem A.","tokens_in":16061,"feed_emoji":"🧩","tokens_out":9560,"duration_ms":105321,"temperature":0.7,"pith_summary":"This paper proves that generalized wreath products built from one-sided subgroup inclusions are nonsofic, even when the ambient group is residually finite and hence sofic. The new condition is infranormality: a subgroup $\\Gamma < G$ is infranormal when the semigroup of elements $g$ with $g\\Gamma g^{-1} \\le \\Gamma$ generates $G$. The main theorem says that if $\\Gamma$ is infranormal but not normal and both $\\Gamma$ and $G$ have Kazhdan's property (T), then $\\left(\\bigoplus_{G/\\Gamma} \\mathbb{Z}/2\\mathbb{Z}\\right)\\rtimes G$ is not sofic. The authors produce explicit residually finite Kazhdan pairs of this kind, so the nonsofic objects are built from groups that are themselves sofic. A companion rigidity statement about fixed-point algebras of probability-measure-preserving actions then gives nonsofic generalized Bernoulli actions and settles an open question about whether every action of a sofic group is sofic.","feed_headline":"Infranormal subgroups force nonsofic wreath products","feed_subtitle":"For explicit residually finite Kazhdan pairs, even generalized Bernoulli actions fail soficity, settling an open question.","key_machinery":"The load-bearing object is the compression semigroup $P_\\Gamma = \\{g \\in G : g\\Gamma g^{-1} \\le \\Gamma\\}$; infranormality means $P_\\Gamma$ generates $G$. The proof proceeds through the expander decomposition of sofic approximations of Kazhdan groups: every sufficiently good approximation decomposes, up to negligible edge edits, into components with a uniform positive Cheeger constant. On these components the authors use the finite cluster groupoid, the groupoid of almost equivariant partial bijections between $\\Gamma$-expander components, whose allowed maps have a strict distance gap: two allowed maps with the same source and target agree either almost everywhere or almost nowhere. Conjugation by a compressor yields a faithful functor between large restrictions of these cluster groupoids. Two bounded-median normalization arguments, one for component vertex mass and one for isotropy order, force the functor to be full on all but negligibly much component weight, which transfers surjectivity to the ultraproduct centralizer. This is what upgrades the diagonal-algebra normalization of Proposition 3.1 to the permutation-centralizer normalization of Theorem 4.1.","core_discovery":"The central claim, stated as Theorem A, is that for countable discrete groups $\\Gamma < G$, if the compression semigroup $P_\\Gamma = \\{g \\in G : g\\Gamma g^{-1} \\le \\Gamma\\}$ generates $G$, if $\\Gamma$ is not normal, and if both $\\Gamma$ and $G$ have Kazhdan's property (T), then the generalized wreath product $\\left(\\bigoplus_{G/\\Gamma} \\mathbb{Z}/2\\mathbb{Z}\\right)\\rtimes G$ is not sofic. The proof establishes a stronger rigidity fact: in every sofic representation $\\sigma$ of $G$, the permutation centralizer of $\\sigma(\\Gamma)$ in the universal sofic group is normalized by $\\sigma(G)$ (Theorem 4.1). A strict compression $t\\Gamma t^{-1} < \\Gamma$ then makes the lamp at the coset $t\\Gamma$ commute with $\\sigma(\\Gamma)$ while a conjugate by some $\\gamma \\in \\Gamma$ does not, contradicting normalization. The same centralizer rigidity implies that $C_G(\\Gamma)$ is normal in $G$ whenever $G$ is sofic (Theorem B). The paper also proves that, under the same hypotheses, the $\\Gamma$-fixed algebra of any sofic p.m.p. action of $G$ is $G$-invariant (Theorem C), so every nontrivial generalized Bernoulli action over $G/\\Gamma$ is nonsofic when $\\Gamma$ is not normal (Corollary D). The explicit pairs in Theorem E, built from elementary matrix groups over polynomial and Laurent polynomial rings, are residually finite and Kazhdan, and they satisfy the infranormality hypothesis.","pith_inferences":["A plausible general template, left implicit in the paper, is that one-sided compression of a rigid subgroup inside a sofic ambient group forces centralizer normalization in any approximation; testing the same mechanism in other rigidity classes, such as groups with strong ergodicity or a spectral gap, could yield further nonsofic wreath products.","The proof uses $\\mathbb{Z}/2\\mathbb{Z}$ mainly through the existence of a nontrivial lamp element with vanishing trace, so a natural extension is that the same conclusion holds with any nontrivial finite group in place of the lamp group.","The paper defines the sofic envelope of $\\Gamma$ as the set of group elements forced to fix all $\\Gamma$-fixed vectors in sofic actions; a direct next question, not settled here, is whether this envelope is exactly the normal closure of $\\Gamma$ or can be strictly larger."],"forward_implications":["For every sofic group $G$ with property (T) and every infranormal Kazhdan subgroup $\\Gamma$, the centralizer $C_G(\\Gamma)$ is normal in $G$.","For the explicit residually finite groups of Theorem E, the coset action $G \\curvearrowright G/\\Gamma$ is nonsofic as an action on a countable set, even though $G$ itself is sofic.","Every nontrivial generalized Bernoulli action over $G/\\Gamma$ is nonsofic, giving a negative answer to the question whether every probability-measure-preserving action of a sofic group is sofic.","The same groups admit a free ergodic strongly ergodic nonsofic p.m.p. action, and hence a nonsofic orbit equivalence relation generated by a free action of a residually finite group.","Because the groups in Theorem E are residually finite, the nonsoficity conclusions apply to groups with explicit finite-quotient approximations, not only to abstract existence arguments."],"supporting_citations":[{"why":"supplies the first nonsofic group and the component-matching mechanism that the paper analyzes and extends to permutation centralizers.","marker":"[15]"},{"why":"provides the structure theorem that sofic approximations of property-(T) groups decompose into uniform expander components after negligible edits.","marker":"[13]"},{"why":"is imported for the finite cluster groupoid and the repair theorem that Lemma 4.2 uses to represent elements of the permutation centralizer by patched total bisections.","marker":"[1]"},{"why":"defines soficity for actions on countable sets and proves the wreath-product closure result whose contrapositive turns Theorem A into nonsoficity of the coset action.","marker":"[11]"},{"why":"defines sofic p.m.p. actions through diagonal-preserving crossed-product embeddings and poses the question answered negatively by Corollary D.","marker":"[16]"},{"why":"supplies property (T) for elementary groups over finitely generated rings, which verifies the hypotheses of the explicit examples.","marker":"[9]"},{"why":"records that property (T) is closed under extensions, used to show the semidirect products in Theorem E have property (T).","marker":"[4]"}],"fun_headline_variants":["Infranormal subgroups force nonsofic wreath products","Nonsofic wreath products from infranormal subgroups","Infranormal + Kazhdan yield nonsofic wreath products","Property (T) and infranormality break soficity","Generalized Bernoulli actions nonsofic over infranormal pairs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported repair theorem that almost-commuting partial bijections between expander components can be repaired, at one jointly chosen scale for all compressors, into a finite cluster groupoid whose arrows exhaust the permutation centralizer; the present paper does not prove that theorem, only the scale bookkeeping around it.","fun_headline_variants_meta":{"raw":{"variants":["Infranormal subgroups force nonsofic wreath products","Nonsofic wreath products from infranormal subgroups","Infranormal + Kazhdan yield nonsofic wreath products","Property (T) and infranormality break soficity","Generalized Bernoulli actions nonsofic over infranormal pairs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000221,"raw_usage":{"total_tokens":1487,"prompt_tokens":1018,"completion_tokens":469,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":382}},"tokens_in":634,"tokens_out":469,"duration_ms":5357,"temperature":1.0,"reasoning_tokens":382,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:08:45.005358+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a sofic approximation of one of the explicit groups in Theorem E in which the centralizer of $\\sigma(\\Gamma)$ fails to be normalized by $\\sigma(G)$, or find a sofic p.m.p. action of that group whose $\\Gamma$-fixed algebra is not $G$-invariant; either example would contradict Theorem C and therefore break the chain leading to Theorem A.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the first nonsofic group and the component-matching mechanism that the paper analyzes and extends to permutation centralizers."},{"cited_title":"On sofic approximations of Property (T) groups","cited_arxiv_id":"1606.04471","evidence_quote":"provides the structure theorem that sofic approximations of property-(T) groups decompose into uniform expander components after negligible edits."},{"cited_title":"Alekseev and A","cited_arxiv_id":null,"evidence_quote":"is imported for the finite cluster groupoid and the repair theorem that Lemma 4.2 uses to represent elements of the permutation centralizer by patched total bisections."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines soficity for actions on countable sets and proves the wreath-product closure result whose contrapositive turns Theorem A into nonsoficity of the coset action."},{"cited_title":"Păunescu,On sofic actions and equivalence relations, J","cited_arxiv_id":null,"evidence_quote":"defines sofic p.m.p. actions through diagonal-preserving crossed-product embeddings and poses the question answered negatively by Corollary D."},{"cited_title":"Ershov and A","cited_arxiv_id":null,"evidence_quote":"supplies property (T) for elementary groups over finitely generated rings, which verifies the hypotheses of the explicit examples."},{"cited_title":"Bekka, P","cited_arxiv_id":null,"evidence_quote":"records that property (T) is closed under extensions, used to show the semidirect products in Theorem E have property (T)."}],"review_version":1}