Nonsofic wreath products of residually finite groups

Andreas Thom; Gabor Kun

REVIEW 2 major objections 4 minor 18 references

Nonsofic wreath products of residually finite groups

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Serious and likely important, but the main theorem is conditional on an unpublished companion preprint: Lemma 4.2's core representation is imported from [1], not proved here. read the letter →

arxiv 2608.06222 v1 pith:YIGQGVOL submitted 2026-08-06 math.GR

Gabor Kun , Andreas Thom This is my paper · ORCID

classification math.GR MSC 20F6922D5537A1537A2046L10

keywords soficgroupsKazhdan'sproperty(T)infranormalsubgroupscompressionsemigroupgeneralizedwreathproductsactionsexpanderdecompositionsresiduallyfinite

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What'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.

The 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.

Soft 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.

Who 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.

Referee Report

2 major / 4 minor

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).

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 (2)

[§4, Lemma 4.2] 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.
[§4, Lemma 4.2(1)] 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].

minor comments (4)

[§1, Abstract and Introduction] The phrase 'breakthrough of OpenAI' is informal and could be replaced by 'the construction of [15]' or 'the result of [15]' throughout the paper.
[§4, Lemma 4.2] 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].
[References] 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.
[§5.2, Proof of Theorem E] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

Main nonsoficity proof rests on Lemma 4.2, whose centralizer-representation part is imported from the same authors' unpublished preprint [1] rather than proved; the surrounding argument has independent content.

self citation load bearing [Section 4, Lemma 4.2(4) and its proof (pages 8–9)]
"(4) Every element of CSU(σ(Γ)) can be represented, up to oU(1) in Hamming distance, by permutations obtained by patching the arrows of total bisections of Cn on Y good n and extending over its negligible complement. Conversely, every sequence of such extended patched bisections represents an element of CSU(σ(Γ)). ... Proof. This is the construction of [1, Proposition 3.3, Lemma 3.4, Definition 4.1 and Proposition 4.5], with the scale chosen diagonally. ... Finally, the two-sided majority argument for almost centralizing permutations, applied componentwise, gives (4)."

Theorem 4.1's proof needs Lemma 4.2(4) twice: to represent an arbitrary centralizer element v as a patched total bisection and to certify that the completed permutation b̃_n lies in CSU(σ(Γ)). The lemma's proof, however, contains no argument for this representation; it says the two-sided majority argument gives (4) and refers to [1, Proposition 4.5]. Since [1] is an unpublished preprint by the second author and a coauthor, the load-bearing bridge between centralizer elements and cluster-groupoid bisections is imported by self-citation rather than established here. If that representation fails, Theorem 4.1 and hence Theorems A and B lose their foundation.

full rationale

The paper does not assume its main conclusions, and the nonsoficity theorems follow from the normalization theorem by a legitimate contradiction argument using the strict compressor. The examples in Theorem E are checked from external property-(T) and residual-finiteness inputs. However, the centralizer normalization (Theorem 4.1) is not self-contained: Lemma 4.2, and especially part (4), is quoted from the same authors' preprint [1] with only scale bookkeeping supplied. This makes the main proof conditional on an unverified same-author citation. That is the only significant circularity-type defect; the rest of the derivation has independent mathematical content, so the score is moderate rather than high.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof introduces many auxiliary numerical parameters (epsilon_n, delta_n, eta_n, q_n, r_n, etc.), but all are chosen to tend to zero along the fixed ultrafilter and are not fitted to data. No empirical constants are fitted, and the central claim depends on the listed prior theorems rather than on newly postulated physical or combinatorial entities. The new mathematical definitions (compression semigroup, infranormal subgroup, sofic Mautner envelope) are tracked as definitions, not as empirical postulates.

assumptions (5)

standard math Exponder decomposition theorem for Kazhdan groups: every sofic representation of a property (T) group can be changed on o(|Y_n|) edges into a disjoint union of components with a uniform positive Cheeger constant.
Invoked in Sections 2, 3, and 4 as the starting point for normalizing the component partitions of the Gamma-action and the ambient G-action. It is attributed to Kun [13, Theorem 1].
domain assumption Cluster groupoid construction: allowed partial bijections between expander components form a finite groupoid with a distance gap, and elements of the centralizer C_{S_U}(sigma(Gamma)) are represented by patched total bisections.
This is the central structural input for Theorem 4.1. The paper quotes it from the same authors' preprint [1] and does not prove the underlying repair theorem or the representation statement; only the joint-scale bookkeeping is given in Lemma 4.2.
standard math Property (T) for EL_r over finitely generated unital rings (Ershov and Jaikin-Zapirain, [9, Theorem 1.1]) and for SL_d(Z) with d at least 3, together with closure of property (T) under extensions.
Used in the proof of Theorem E to show that both Gamma and G are Kazhdan, which is a hypothesis of Theorems A, B, and C.
standard math Soficity of p.m.p. actions passes to factors, and for free actions it is equivalent to soficity of the orbit equivalence relation (Paunescu, [16, Proposition 1.15]).
Used in Corollary 3.2 to pass from a nonsofic Bernoulli action to a nonsofic free ergodic action and then to a nonsofic equivalence relation.
standard math Every ergodic p.m.p. action of a Kazhdan group is strongly ergodic (Connes and Weiss, [5]).
Used in Corollary 3.2 to obtain strong ergodicity of the constructed free ergodic nonsofic action.

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Cite this review

Pith. "Pith review of Nonsofic wreath products of residually finite groups." pith.science (2026). https://pith.science/paper/YIGQGVOL

@misc{pith2026260806222,
  author       = {Pith},
  title        = {Pith review of: Nonsofic wreath products of residually finite groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YIGQGVOL}},
  note         = {Machine review of arXiv:2608.06222}
}

abstract

This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $\Gamma<G$ be such that $\{g\in G:g\Gamma g^{-1}\leq\Gamma\}$ generates $G$ as a group, and suppose that both $\Gamma$ and $G$ have property $(T)$. If $\Gamma$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ is nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Alekseev and A

V. Alekseev and A. Thom,Centralizers of sofic approximations of Kazhdan groups, submitted to arXiv, 2026

work page 2026

[2] [13]

On sofic approximations of Property (T) groups

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work page Pith review arXiv 2019

[3] [14]

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G. Kun and A. Thom,Inapproximability of actions and Kazhdan’s property(T ), arXiv:1901.03963v3 [math.GR], 2026

work page Pith review arXiv 1901

[4] [15]

OpenAI,A counterexample to the soficity conjecture, inTen advances in mathematics and theoretical computer science, Chapter 3, 77–93, 2026,https://cdn.openai.com/pdf/ten-proofs-oai.pdf

work page 2026

[5] [2]

Alekseev and A

V. Alekseev and A. Thom,Remarks on approximability and stability for groups, arXiv:2512.15494v1 [math.GR], 2025

work page arXiv 2025

[6] [3]

Bader, A

U. Bader, A. Furman, A. Gorodnik and B. Weiss,Rigidity of group actions on homogeneous spaces, III, Duke Math. J.164(2015), no. 1, 115–155

work page 2015

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Bekka, P

B. Bekka, P. de la Harpe and A. Valette,Kazhdan’s property(T ), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008

work page 2008

[8] [5]

Connes and B

A. Connes and B. Weiss,Property( T )and asymptotically invariant sequences, Israel J. Math.37 (1980), no. 3, 209–210. 16 GÁBOR KUN AND ANDREAS THOM

work page 1980