arXiv:2608.05362v1

Centralizers of sofic approximations of Kazhdan groups

Vadim Alekseev, Andreas Thom

math.GR20F6520F6920F0537A1546L10

Paper metadata

Original submission: August 5, 2026
Claimed MSC: 20F65, 20F69, 20F05, 37A15, 46L10
arXiv categories: math.GR
Paper license: CC BY 4.0 ↗
Comments: 19 pages, no figures

Abstract

We prove that a Kazhdan group admitting a sofic embedding into a metric ultraproduct of symmetric groups with a centralizer that acts ergodically on the associated Loeb probability space is locally embeddable in finite groups (LEF). In particular, every finitely presented Kazhdan group admitting such an embedding is residually finite. The main technical theorem says that the centralizer of a sofic embedding of a Kazhdan group is itself a metric ultraproduct of permutation groups.

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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.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The finite-level description of the centralizer of a Kazhdan sofic embedding and the deduction that an ergodic centralizer forces the group to be LEF, and residually finite when finitely presented, are correct.

Theorem 3.1Correct

Centralizer rigidity after an essentially equivalent replacement

Pages 6 and 16–17 · Theorem 3.1 and its proof · arXiv:2608.05362v1

After Kun's decomposition, the finite model is a disjoint union of uniformly expanding components. Clusters of almost equivariant partial bijections form finite groupoids, and their finite full groups act by uniformly approximate permutations. Proposition 4.5 proves both directions needed for centralizer equality: every full-group sequence asymptotically commutes with the sofic image, and every ultraproduct centralizer element is approximated by such a sequence. Uniform flexible stability for finite groups replaces these approximate actions by genuine finite permutation actions after adding $o(|X_n|)$ points. Hence the essentially equivalent embedding satisfies $C_{\prod_{\mathcal U}\operatorname{Sym}(Y_n)}(\pi'(G))=\prod_{\mathcal U}A_n$ for finite $A_n\leq\operatorname{Sym}(Y_n)$ .

Full paper, version 1 ↗

Theorem ACorrect

Ergodic centralizer implies LEF and residual finiteness

Pages 2 and 17 · Theorem A and its proof · arXiv:2608.05362v1

Theorem 3.1 realizes the centralizer as $\prod_{\mathcal U}A_n$ . Ergodicity of its Loeb action forces an $A_n$ -orbit occupying asymptotically all of the model. After restricting to those orbits, $A_n$ acts transitively, so its exact permutation centralizer is $N_{A_n}(L_n)/L_n$ acting freely. Proposition 2.7 places $G$ in the corresponding algebraic ultraproduct of finite groups, which is exactly the LEF condition for a finitely generated group. A finite presentation converts sufficiently accurate local embeddings into genuine separating finite quotients, giving residual finiteness.

Full paper, version 1 ↗

Corollary 5.2Correct

Consequence for the Hayes–Kunnawalkam Elayavalli conjecture

Page 17 · Corollary 5.2 · arXiv:2608.05362v1

The conjectured embedding has exactly the ergodic-centralizer hypothesis of Theorem A, so the LEF conclusion, and the residual-finiteness conclusion in the finitely presented case, follow by direct substitution.

02Proofs6 reported findingsCorrect

The central proofs are correct and complete. The transitive-centralizer lemma, LEF reduction, expander repair, finite cluster-groupoid construction, exhaustion of the ultraproduct centralizer, and flexible-stability correction all support their stated uses. Two local notation defects are listed below and do not affect the argument.

Propositions 2.7 and 2.9Correct and complete

Transitive finite centralizers and the LEF reduction

Pages 5–6 · Propositions 2.7 and 2.9 · arXiv:2608.05362v1

For $X_n=A_n/L_n$ , uniform almost-commutation with the transitive $A_n$ -action makes the majority value of $g^{-1}\sigma_n(gL_n)$ unique and forces it to be a coset in $N_{A_n}(L_n)/L_n$ . Thus the ultraproduct centralizer is the ultraproduct of the exact finite centralizers. Their right actions on $A_n/L_n$ are free, so distinct elements have Hamming distance one. The resulting metric ultraproduct is therefore the algebraic ultraproduct, and every finite multiplication table in its finitely generated subgroup is realized in one finite factor.

Theorem 2.5 and Proposition 3.3Correct and complete

Expander decomposition and repair of partial intertwiners

Pages 4 and 7–9 · Theorem 2.5 and Proposition 3.3 · arXiv:2608.05362v1

Kun's theorem supplies an essentially equivalent union of components with one uniform Cheeger constant. For a partial intertwiner $b:Y\dashrightarrow Z$ , its graph in the diagonal $S$ -labelled product has boundary bounded by $\Delta_S(b)|Y|$ . Removing vertices with incorrect local models costs only the chosen $\delta$ -fraction. The cited Kazhdan repair proposition produces a nearby set with small boundary; expansion in the two coordinates then controls empty and multiple rows and columns, allowing it to be corrected to a partial bijection with arbitrarily small source, range, and equivariance defects. The distance estimate retains a constant multiple of the original defect plus the prescribed tolerance.

Kun–Thom, Proposition 3.3 ↗

Lemma 4.2 and Proposition 4.5Correct and complete

The cluster groupoid is well defined and exhausts the centralizer

Pages 11–15 · Lemmas 4.2, 4.4 and Proposition 4.5 · arXiv:2608.05362v1

Expansion gives the required distance gap: two allowed partial maps are either $q_n$ -close or at least $1-q_n$ apart. This gap makes improvement-based composition independent of representatives and proves the inverse and associativity axioms. For a centralizing sequence $\sigma_n$ , the two-sided majority decomposition assigns almost every source component an injective majority target. Markov bounds discard only $o(|X_n|)$ component weight; the repair theorem supplies an allowed arrow on every remaining component, and injectivity lets these arrows be completed within each groupoid component to a total bisection. The resulting permutation is $o(1)$ -close to $\sigma_n$ , proving the nontrivial reverse inclusion.

Proposition 5.1Correct and complete

Uniform flexible stability produces genuine finite actions

Pages 16–17 · Proposition 5.1 · arXiv:2608.05362v1

Proposition 4.5 supplies maps from the finite groups $F_n=[[C_n]]$ with uniform multiplicative defect tending to zero. Becker–Chapman's theorem applies because every finite $F_n$ is amenable and gives a homomorphism on a set enlarged by only an asymptotically negligible fraction, uniformly close on all of $F_n$ . Uniform asymptotic commutation proves one inclusion in the new centralizer equality. Conversely, a centralizer representative can be altered on the negligible added set to preserve $X_n$ , then Proposition 4.5(c) approximates its restriction by a full-group element; uniform closeness of the corrected action gives the other inclusion.

Becker–Chapman, Theorem 1.2 ↗

Definition 2.1Typo

The multiplicativity variables are omitted after “for all”

Page 3 · Definition 2.1 · arXiv:2608.05362v1

The printed definition says “for all” and then displays $d_H(\theta_n(\lambda\mu),\theta_n(\lambda)\theta_n(\mu))\to0$ without supplying the quantifier. It must read “for all $\lambda,\mu\in\Lambda$ .” The variables, domain, and intended quantification are uniquely fixed by the displayed formula and by the equivalent free-group formulation immediately below, so this omission does not affect any later argument.

Proof of Theorem ATypo

The essential-equivalence citation omits one item number

Page 17 · final proof, second paragraph · arXiv:2608.05362v1

The proof cites “Lemma 2.4(ii)” for conjugacy of the embeddings, identification of centralizers, and preservation of ergodicity. The last two conclusions are Lemma 2.4(iii), so the citation should be “Lemma 2.4(ii)–(iii).” Both items were stated and proved available earlier, and the argument uses them correctly.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.05362v1
Authors listed: Vadim Alekseev, Andreas Thom
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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Centralizers of sofic approximations of Kazhdan groups

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Centralizer rigidity after an essentially equivalent replacement

Ergodic centralizer implies LEF and residual finiteness

Consequence for the Hayes–Kunnawalkam Elayavalli conjecture

Transitive finite centralizers and the LEF reduction

Expander decomposition and repair of partial intertwiners

The cluster groupoid is well defined and exhausts the centralizer

Uniform flexible stability produces genuine finite actions

The multiplicativity variables are omitted after “for all”

The essential-equivalence citation omits one item number

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