arXiv:2608.06222v1

Nonsofic wreath products of residually finite groups

Gabor Kun, Andreas Thom

math.GR20F6922D5537A1537A2046L10

Paper metadata

Original submission: August 6, 2026
Claimed MSC: 20F69, 22D55, 37A15, 37A20, 46L10
arXiv categories: math.GR
Paper license: CC BY 4.0 ↗
Comments: 16 pages, no figures

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 $Γ<G$ be such that $\{g\in G:gΓg^{-1}\leqΓ\}$ generates $G$ as a group, and suppose that both $Γ$ and $G$ have property $(T)$ . If $Γ$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/Γ}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ 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.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The nonsoficity, centralizer-normality, fixed-algebra, nonsofic-action, and explicit-example results are correct under the stated Kazhdan and infranormality hypotheses.

Theorems A and BCorrect

Infranormal Kazhdan subgroups force the stated sofic obstructions

Pages 2 and 14 · Theorems A–B and their proofs · arXiv:2608.06222v1

Theorem 4.1 normalizes the permutation centralizer of $\sigma(\Gamma)$ by $\sigma(G)$ . In the generalized wreath product, the lamp at $\Gamma$ centralizes $\Gamma$ , so every conjugate lamp would have to do so. A strict compressor $t\Gamma t^{-1}<\Gamma$ supplies $\gamma\in\Gamma\setminus t\Gamma t^{-1}$ , for which the lamp at $t\Gamma$ is moved by $\gamma$ , giving the required contradiction to soficity. Intersecting the normalized ultraproduct centralizer with the injective copy of $G$ gives $\sigma(C_G(\Gamma))=\sigma(G)\cap C_{S_{\mathcal U}}(\sigma(\Gamma)),$ and therefore proves the normality asserted in Theorem B.

Theorem C and Corollary DCorrect

The fixed algebra is invariant and the coset Bernoulli action is nonsofic

Pages 2 and 7 · Theorem C, Corollary D, and their proofs · arXiv:2608.06222v1

The expander decompositions for the restrictions to $G$ and $\Gamma$ meet Proposition 3.1, so the $\Gamma$ -fixed diagonal algebra is normalized by $\sigma(G)$ . Intersecting it with the embedded copy of $L^\infty(X)$ proves $G$ -invariance of $L^\infty(X)^\Gamma$ . In the coset Bernoulli action, the coordinate projection at $\Gamma$ is $\Gamma$ -fixed, while its translate at $g\Gamma$ is $\Gamma$ -fixed exactly when $\Gamma\leq g\Gamma g^{-1}$ . Applying this to both $g$ and $g^{-1}$ would make $\Gamma$ normal, so a nontrivial action over $G/\Gamma$ cannot be sofic when $\Gamma$ is not normal.

Theorem ECorrect

The Laurent-polynomial pair has every required property

Pages 2 and 15–16 · Theorem E and Section 5.2 · arXiv:2608.06222v1

For $r,d\geq3$ , Ershov–Jaikin-Zapirain gives property $(T)$ for the elementary groups over both finitely generated rings, and the semidirect product retains property $(T)$ . Positive elementary matrices in $\operatorname{SL}_d(\mathbb Z)$ compress $\Gamma=\operatorname{EL}_r(\mathbb F_q[\mathbb N^d])$ ; the compressor $I+E_{12}$ is strict. Their group generates $\operatorname{SL}_d(\mathbb Z)$ , and transitivity on primitive exponent vectors supplies every Laurent monomial elementary matrix, proving infranormality. Reduction of exponent vectors and matrices modulo a suitably chosen $m$ separates every nonidentity element, proving residual finiteness.

Ershov–Jaikin-Zapirain, Theorem 1.1 ↗

02Proofs4 reported findingsCorrect

The two expander-component arguments, cluster-groupoid centralizer normalization, lamp contradiction, fixed-algebra argument, and explicit algebraic construction are correct and complete. The cited expander and cluster inputs were checked against their primary sources.

Proposition 3.1Correct and complete

Component medians upgrade one-sided compression to diagonal normalization

Pages 5–7 · Proposition 3.1 · arXiv:2608.06222v1

A compressor gives an almost matching from the $\Gamma$ -component partition to its conjugate. On each ambient expander component, the bounded median argument makes the component-size ratio asymptotically constant, and ambient expansion forces the exceptional mass to vanish. Lemma 2.3 then identifies the two component diagonal algebras. Repeating this for finitely many compressors that generate $G$ proves normalization by all of $G$ . The property- $(T)$ expander decomposition invoked here matches Kun's Theorem 1.

Kun, On sofic approximations of Property (T) groups ↗

Theorem 4.1Correct and complete

The cluster-groupoid construction normalizes the full permutation centralizer

Pages 8–14 · Lemmas 4.2–4.4 and proof of Theorem 4.1 · arXiv:2608.06222v1

The cited cluster construction represents centralizer elements by total bisections and supplies a distance gap and faithful one-sided functor on conull component families. The first median comparison makes orbit counts asymptotically equal; the second makes the isotropy injection have integer index below two, hence index one. This turns the functor into a bijection on the required Hom-sets, so the transported partial bisection can be completed within each cluster at vanishing cost. The reverse inclusion follows directly from $t\Gamma t^{-1}\leq\Gamma$ . These steps establish equality under conjugation by every generating compressor and therefore normalization by $G$ .

Alekseev–Thom, Centralizers of sofic approximations of Kazhdan groups ↗

Theorems A–DCorrect and complete

The algebraic and dynamical deductions use the normalization results correctly

Pages 7 and 14 · proofs of Theorem C, Corollary D, Theorems A–B · arXiv:2608.06222v1

The fixed-algebra intersection is normalized by both required factors; distinct Bernoulli coordinates remain distinct for every nontrivial base; and applying the fixedness inclusion to $g$ and $g^{-1}$ yields equality of conjugate subgroups. In the wreath-product argument, the chosen $\gamma$ moves the lamp because $\gamma t\Gamma\neq t\Gamma$ , while injectivity of the sofic representation preserves the nontrivial lamp product. No additional normality or finiteness assumption is used.

Theorem ECorrect and complete

Kazhdan property, strict compression, generation, and residual finiteness are all verified

Pages 15–16 · proof of Theorem E · arXiv:2608.06222v1

The property- $(T)$ input applies because both coefficient rings are finitely generated and $r\geq3$ . Strictness is witnessed by an elementary matrix whose coefficient is absent from the compressed polynomial subring. Completing a primitive exponent vector to an oriented integral basis supplies every positive and negative Laurent monomial after conjugation, and elementary additivity supplies every Laurent polynomial. Finally, reduction modulo $m$ is compatible with the monomial action, and choosing $m$ larger than all coordinate differences in a nonzero finite support prevents cancellation. This completes all four assertions.

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Paper: arXiv:2608.06222v1
Authors listed: Gabor Kun, Andreas Thom
Audit date: August 18, 2026

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Nonsofic wreath products of residually finite groups

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Infranormal Kazhdan subgroups force the stated sofic obstructions

The fixed algebra is invariant and the coset Bernoulli action is nonsofic

The Laurent-polynomial pair has every required property

Component medians upgrade one-sided compression to diagonal normalization

The cluster-groupoid construction normalizes the full permutation centralizer

The algebraic and dynamical deductions use the normalization results correctly

Kazhdan property, strict compression, generation, and residual finiteness are all verified

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