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On automorphism groups of metric reduced products of symmetric groups

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Assuming the open coloring axiom and Martin's axiom for sigma-linked posets, every isomorphism between metric reduced products of symmetric groups is trivial: up to inner conjugation it is a coordinate permutation induced by an almost…

desk verdict Substantial and likely correct paper, but the proof of Theorem 3.5 has a definability gap that needs fixing. read the letter →

arxiv 2412.10802 v1 pith:NIYCUF3L submitted 2024-12-14 math.GR math.LO

Ben De Bondt , Andreas Thom This is my paper · ORCID
classification math.GRmath.LO MSC 20B3003C20
keywords metricreducedproductssymmetricgroupsHammingouterautomorphismgroupopencoloringaxiomMartin'ssoficrigidity
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

Metric reduced products of finite symmetric groups, equipped with the normalized Hamming metric, are the universal sofic groups: every sofic group embeds into one of them. The paper studies what happens to their isomorphisms under two forcing axioms, OCA and Martin's axiom for sigma-linked posets. Its central claim is that, under those axioms, every isomorphism between two such products is trivial in a precise sense: it is, up to conjugation in each coordinate, just a permutation of the coordinates that asymptotically preserves the coordinate sizes. This would give a complete description of the outer automorphism group of each such product and would contrast sharply with the situation under the continuum hypothesis, where these groups are extremely flexible. The same results also yield a concrete count of the isomorphism classes of these groups.

What carries the argument

The central objects are the metric reduced product Sym[(k_n)_n] and the notion of a trivial isomorphism, built from an almost permutation f of the natural numbers together with the coordinatewise cut/lift operation and inner conjugation. The argument runs through four linked mechanisms. First, an Ulam-stability theorem for approximate homomorphisms from Sym(k_n) to Sym(l_n) forces any product-form isomorphism to be inner and to satisfy lim_n k_n/l_n = 1. Second, the proof attaches to every isomorphism an automorphism $\theta$ of the Stone-Cech boundary of the natural numbers by defining the structure ([0,1]^N/Fin, 0, 1, <=, not, o-plus) inside the group via involution conjugacy classes and the normalized Hamming distance to the identity. Third, OCA converts that boundary automorphism into an almost permutation. Fourth, the metric lifting theorem converts the resulting pseudometric-preserving isomorphism into product form, where the stability theorem applies.

What would settle it

In Sym(6), the uniqueness used to define not D1(a) fails: for a=(12)(34)(56) and b=(13)(24), the defining conditions are satisfied by b with D1(b)=2/3 and D1(ab)=1, not with D1(b)=1. This refutes the definability lemma behind Theorem 3.5, so the proof as written has a concrete gap.

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Extended reading notes

Core claim

The paper's main theorem states that, assuming OCA and MA_alpha1($\sigma$-linked), every isomorphism phi from Sym[(k_n)_n] to Sym[(l_n)_n] is trivial. Concretely, there is an almost permutation f of the natural numbers with lim_n k_{f(n)}/l_n = 1 and an element $\sigma$ of Sym[(l_n)_n] such that phi maps each element (a_n)_n to (sigma_n (a_{f(n)} updownarrow Sym(l_n)) $sigma_n^{{-1}}$)_n, where updownarrow denotes cutting a permutation to a smaller set or lifting it by adding fixed points. The paper derives two consequences: the outer automorphism group of Sym[(k_n)_n] is isomorphic to the group of almost permutations f with lim_n k_{f(n)}/k_n = 1, and the groups Sym[(k_n)_n] and Sym[(l_n)_n] are isomorphic exactly when the two sequences are asymptotic almost rearrangements of each other. It also proves that, under the same axioms, elementary equivalence does not imply isomorphism, while under the continuum hypothesis the groups are elementarily equivalent exactly when they are isomorphic.

Load-bearing premise

The proof rests on the claim that the operation turning an involution's support fraction into one minus that fraction is uniquely definable from the group structure; that uniqueness fails in concrete finite symmetric groups.

Editorial extensions

If this is right

  • The outer automorphism group of Sym[(k_n)_n] is exactly the group of almost permutations of N that asymptotically preserve the sequence k_n.
  • Two such metric reduced products are isomorphic exactly when their defining sequences are asymptotic almost rearrangements of each other.
  • If the logarithms of the coordinate sizes are separated by a positive lower bound, then Sym[(k_n)_n] is complete: every automorphism is inner.
  • Under OCA and MA_alpha1(sigma-linked), there are elementarily equivalent but non-isomorphic groups Sym[(k_n)_n], a phenomenon that cannot occur under the continuum hypothesis.
  • There are exactly continuum-many isomorphism classes of metric reduced products of symmetric groups.

Reading between the lines

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

  • The same proof route, with a suitable Ulam-stability result, is expected to transfer the triviality theorem to unitary groups with normalized trace norm, yielding rigidity for the unitary group of the corona algebra; the paper lists this as a natural next step.
  • The dichotomy between the forcing-axiom universe and the continuum-hypothesis universe suggests a model-theoretic signature: under CH, isomorphism coincides with elementary equivalence, whereas under OCA/MA it is a purely combinatorial condition on the defining sequences.
  • The rigidity pattern may extend to other reduced products of finite metric groups with bi-invariant metrics, where the same combination of Ulam stability, Stone-Cech boundary automorphisms, and metric lifting could apply.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies isomorphisms between metric reduced products Sym[(k_n)_n] of finite symmetric groups with the normalized Hamming metric. Assuming OCA and MA_ℵ1(σ-linked), it claims that every isomorphism between two such groups is trivial in the sense of Definition 2.4: up to conjugation by an element of the target group, it is the coordinate permutation ψ_f associated with an almost permutation f with lim_n k_{f(n)}/l_n = 1. From this it derives a description of the outer automorphism group as Out((k_n)_n), a criterion for completeness, a dichotomy with the CH behavior, and consequences for elementary equivalence and non-isomorphism. The proof strategy is: Theorem 3.5 shows that any isomorphism induces an automorphism θ of P(N)/Fin preserving zero-distance relations; [5, Theorem 1] trivializes θ under OCA; the metric lifting theorem [6] turns the resulting coordinate-respecting map into product form; and the Ulam stability result Theorem 3.2 forces the product form to be inner. The manuscript also proves that all isomorphisms are automatically isometric.

Significance. If the main theorem is correct, this is a substantial rigidity result for universal sofic groups under natural forcing axioms: isomorphism types are completely controlled by asymptotic rearrangements of the defining sequences, and the automorphism group is a split extension with an explicitly computed outer group. The architecture is attractive and reusable: it cleanly packages the black boxes [5] and [6], reduces the product-form case to Ulam stability, and produces explicit, checkable corollaries. The paper contains no fitted parameters and the central claim is not definitionally circular. However, the proof of Theorem 3.5 contains a genuine gap in the definability argument for the complement operation, and since Theorem 3.9 and its corollaries rely on Theorem 3.5, the main rigidity statement is not established as written. The gap appears local and likely repairable through the order-definability part of the same theorem, but the repair is not present in the manuscript.

major comments (2)
  1. [Section 3.2, proof of Theorem 3.5] The uniqueness claim used to define ¬ is false. In a block of six points, let a=(12)(34)(56) and consider b=id and b=(13)(24). Both commute with a, and in both cases D1(a)⊕D1(b)=1 and D1(ab)=1, yet D1(b)=0 in the first case and 2/3 in the second. Taking such blocks for a sequence (k_n) of multiples of 6 tending to infinity gives two distinct involutions b in Sym[(k_n)_n] with different D1-values that satisfy the stated conditions. Consequently the sentence "Finally, for an involution a ... uniquely determined ..." does not establish definability of ¬, and the subsequent assertion that φ induces an automorphism of ([0,1]^N/Fin,0,1,≤,¬,⊕) is unsupported.
  2. [Theorem 3.9 and Corollaries 3.11, 3.13] The gap in Theorem 3.5 is load-bearing: property (1) and the existence of θ are used in Theorem 3.9 to apply [5, Theorem 1] and [6], and the corollaries depend on Theorem 3.9. A repair is needed. The order-definability part of the proof may suffice: if one proves directly from the definability of ≤ (and 0,1) that φ induces an automorphism of the lattice C(∂βN,[0,1]) fixing 0 and 1, and that every such automorphism is composition with a homeomorphism, then the rest of the argument can go through. The manuscript currently cites [4] and [16] only for the full expanded structure, so it does not supply this lattice-only argument.
minor comments (3)
  1. [Abstract and general text] There are several typographical and spacing issues, e.g., "assu ming" in the abstract and the spacing in "Todorˇcevi´c"; a proofreading pass is recommended.
  2. [Proof of Corollary 3.7] The notation Sym[(k^ν_n)_n]/U is ambiguous, since it could be read as a quotient of the reduced product by an ultrafilter; the earlier notation ∏_n Sym(k_n)/U in Theorem 3.5(2) is clearer and should be used consistently.
  3. [Proof of Theorem 3.5] The step "Basic combinatorics shows that b is a product of two conjugates of a" should be expanded or given a reference; it is not immediate that parity obstructions disappear for equality in the metric reduced product quotient, and this point is used to pass from involutions to arbitrary elements.

Circularity Check

0 steps flagged · score 1.0 of 10

No definitional circularity; the main theorem is derived from independent lifting and stability theorems. The paper relies on same-author prior work, but those results do not contain the target theorem and are used as black boxes.

full rationale

The derivation of Theorem 3.9 does not fit a parameter and then relabel it as a prediction. Triviality is a target classification, not an input: an arbitrary isomorphism φ is shown to be ψ_f up to conjugation, with the asymptotic condition lim k_{f(n)}/l_n=1 obtained from Lemma 3.4 rather than assumed. The combinatorial group Out((k_n)_n) is defined independently of Aut(Sym[(k_n)_n]) and Corollary 3.13 proves an isomorphism instead of taking one as a definition. The cited [5] and [6] are prior works with overlapping authors, but they are invoked for results whose statements do not include this paper's main theorem: [5, Theorem 1] is the OCA rigidity of automorphisms of P(N)/Fin, and the metric lifting theorem of [6] is a general coordinate-respecting-to-product-form statement. Under the stated review rules these are independent support and do not raise circularity above a minor self-citation level. The disputed uniqueness claim for ¬ in Theorem 3.5, namely that the conditions on a commuting involution b determine D1(b), is a genuine correctness gap in a definability argument, not a circularity: it concerns the validity of a lemma, not the importation of the conclusion into the premises. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on two explicit set-theoretic axioms (OCA and MA_alpha1(sigma-linked)) and on several prior theorems used as black boxes. The Becker-Chapman stability result and Kaplansky's lattice theorem are standard external benchmarks. The main theorems of [5] and [6] are prior results of the same research group; they are general lifting and trivial-isomorphism theorems whose assumptions do not include the target conclusion, so their use is not circular. There are no fitted parameters and no invented entities.

assumptions (6)
  • domain assumption Open Coloring Axiom (OCA) and Martin's axiom MA_alpha1(sigma-linked)
    The main theorem is proved under these two forcing axioms, which are consistent with ZFC and consequences of PFA. They are used in Section 3.3 (Theorem 3.9) to ensure theta is induced by an almost permutation (via [5]) and that the composed map is of product form (via [6]).
  • domain assumption Becker-Chapman Ulam stability theorem ([2, Theorem 1.2])
    Used to prove Theorem 3.2, the stability result for almost surjective almost homomorphisms between finite symmetric groups. It provides the universal constant c.
  • domain assumption Theorem 1 of [5] (OCA implies trivial isomorphisms between reduced products of countable structures are induced by almost permutations)
    Used in Section 3.3 to convert the autohomeomorphism theta into an almost permutation f. This is a prior result by the same research group.
  • domain assumption Main theorem of [6] (metric lifting theorem)
    Used in Section 3.3 to conclude that the composed isomorphism hat-phi respecting d_S for all S is of product form. This is a prior result by De Bondt and Vignati.
  • standard math Classification of permutation representations of Sym(n) for n >= 7 on sets of size < 2n
    Used in the proof of Theorem 3.2 to conclude that an almost homomorphism is close to a conjugation. This is a classical fact.
  • standard math Kaplansky's theorem that automorphisms of C(X,[0,1]) with pointwise order are induced by homeomorphisms of X
    Used in Theorem 3.5 to pass from the induced automorphism of [0,1]^N/Fin to an autohomeomorphism of the Stone-Cech boundary. Cited as [16].

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Pith. "Pith review of On automorphism groups of metric reduced products of symmetric groups." pith.science (2026). https://pith.science/paper/NIYCUF3L

@misc{pith2026241210802,
  author       = {Pith},
  title        = {Pith review of: On automorphism groups of metric reduced products of symmetric groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIYCUF3L}},
  note         = {Machine review of arXiv:2412.10802}
}
abstract

We study isomorphisms between metric reduced products of symmetric groups with the normalized Hamming metric assuming the open coloring axiom $\mathsf{OCA}$ and Martin's axiom for $\sigma$-linked posets.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Trace-norm rigidity for reduced products of unitary groups and matrix algebras

    math.OA 2026-07 conditional novelty 7.0 of 10

    Under OCA+MA, isomorphisms of tracial reduced products of unitary groups or matrix algebras reduce to almost permutations of coordinates plus coordinatewise automorphisms, with asymptotic dimension matching.

Reference graph

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