arxiv: 2604.19174 · v2 · submitted 2026-04-21 · 🧮 math.GR

Recognition: unknown

On minimal non-sofic and ω-non-sofic groups

K{\i}van\c{c} Ersoy

Pith reviewed 2026-05-10 01:29 UTC · model grok-4.3

classification 🧮 math.GR

keywords non-sofic groupsminimal non-sofic groupsω-non-sofic groupsresidually finite groupscentral extensionssimple groupsexistentially closed groups

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The pith

If non-sofic groups exist, minimal non-sofic groups with a finitely generated residually finite maximal normal subgroup are perfect central extensions of finitely generated non-amenable simple groups.

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

The paper assumes that at least one non-sofic group exists and then derives structural restrictions on two new classes of such groups. It focuses first on minimal non-sofic groups, which fail to be sofic yet have every proper quotient sofic. For any such group possessing a finitely generated residually finite maximal normal subgroup, the subgroup must sit in the center and the whole group must be a perfect central extension of a finitely generated non-amenable simple group. The paper further shows that every locally graded non-sofic group is omega-non-sofic, meaning it contains finitely generated non-sofic subgroups whose normal subgroups form strictly decreasing chains with nontrivial intersection inside the profinite residual. Using existentially closed groups, it concludes that the mere existence of one non-sofic group forces the existence of countable existentially closed non-sofic examples whose centralizers form a dense chain of non-sofic subgroups ordered like the rationals, as well as non-sofic groups of unbounded exponent.

Core claim

If G is a minimal non-sofic group and M is a finitely generated residually finite maximal normal subgroup of G, then M is central and G is a perfect central extension of a finitely generated non-amenable simple group. Locally graded non-sofic groups are necessarily omega-non-sofic: they contain finitely generated non-sofic subgroups admitting strictly decreasing chains of finitely generated normal subgroups whose intersection is nontrivial and lies in the profinite residual. The existence of a non-sofic group implies the existence of a countable existentially closed non-sofic group whose centralizers form a densely ordered chain of non-sofic subgroups of order type (Q, <=), and also implies,

What carries the argument

minimal non-sofic group (a non-sofic group whose every proper quotient is sofic), together with analysis of its finitely generated residually finite maximal normal subgroups

If this is right

Any locally graded non-sofic group contains a finitely generated non-sofic subgroup with a strictly decreasing chain of finitely generated normal subgroups whose intersection is nontrivial and contained in the profinite residual.
If any non-sofic group exists then there exists a countable existentially closed non-sofic group whose centralizers form a densely ordered chain of non-sofic subgroups of rational order type.
If any non-sofic group exists then there exists a non-sofic group of unbounded exponent.
The class of omega-non-sofic groups is nonempty whenever the class of non-sofic groups is nonempty.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The structural theorem reduces the problem of locating minimal non-sofic groups to the task of locating non-sofic perfect central extensions of finitely generated non-amenable simple groups.
The dense rational chain of centralizers shows that non-sofic behavior, if it occurs, can be embedded inside groups with highly ordered subgroup lattices.
The results tie the existence question for non-sofic groups to the existence of non-amenable simple groups that admit non-sofic central extensions.

Load-bearing premise

At least one non-sofic group exists.

What would settle it

An explicit minimal non-sofic group that possesses a finitely generated residually finite maximal normal subgroup failing to be central would falsify the main structural claim.

read the original abstract

We investigate structural properties of non-sofic groups, assuming that such groups exist. We introduce and study two classes: minimal non-sofic groups and $\omega$-non-sofic groups. For minimal non-sofic groups, we establish strong structural restrictions. In particular, we show that if $G$ is a minimal non-sofic group and $M$ is a finitely generated residually finite maximal normal subgroup of $G$, then $M$ is central and $G$ is a perfect central extension of a finitely generated non-amenable simple group. On the other hand, we show that locally graded non-sofic groups are necessarily $\omega$-non-sofic. More precisely, such groups contain finitely generated non-sofic subgroups admitting strictly decreasing chains of finitely generated normal subgroups whose intersection is nontrivial and lies in the profinite residual. Finally, using results on existentially closed groups, we prove that the existence of a non-sofic group implies the existence of a countable existentially closed non-sofic group whose centralizers form a densely ordered chain of non-sofic subgroups of order type $(\mathbb{Q},\leq)$. In particular, we show that if a non-sofic group exists, then the class of $\omega$-non-sofic groups is non-empty. Moreover, we prove that the existence of a non-sofic group implies the existence of a non-sofic group of unbounded exponent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines minimal non-sofic and ω-non-sofic groups then derives their forced structural features under the assumption that non-sofic groups exist.

read the letter

The paper introduces minimal non-sofic groups and ω-non-sofic groups, then works out what their internal structure must look like if any non-sofic groups turn out to exist. For a minimal non-sofic group with a finitely generated residually finite maximal normal subgroup, that subgroup is central and the quotient is a finitely generated non-amenable simple group. Locally graded non-sofic groups are shown to be ω-non-sofic, with strictly decreasing chains of normal subgroups whose intersection sits in the profinite residual. Existence of one non-sofic group is also shown to imply countable existentially closed non-sofic examples whose centralizers form a dense chain of order type (Q, ≤), plus non-sofic groups of unbounded exponent. These definitions and conditional statements are new. The arguments rely on standard facts about residual finiteness, profinite residuals, and existentially closed groups, and they stay explicitly conditional without hidden steps. The writing keeps the logic clear and the claims proportionate to the hypothesis. The main limitation is that everything remains hypothetical. No non-sofic group is known, so the results organize possible consequences rather than give concrete examples or settle the soficity question. The proofs appear routine once the definitions are fixed, with no visible circularity or unsupported moves. This work is for specialists already engaged with the soficity problem or related questions in geometric group theory. A reader trying to constrain possible constructions or models of non-sofic groups could find the restrictions useful. It deserves a serious referee because the new classes and their structural links are substantive enough to warrant expert scrutiny, even though the theorems stay conditional on an open assumption.

Referee Report

0 major / 2 minor

Summary. Assuming the existence of non-sofic groups, the paper introduces minimal non-sofic groups and ω-non-sofic groups. It proves that if G is minimal non-sofic with a finitely generated residually finite maximal normal subgroup M, then M is central and G is a perfect central extension of a finitely generated non-amenable simple group. It shows locally graded non-sofic groups are ω-non-sofic, containing finitely generated non-sofic subgroups with strictly decreasing chains of finitely generated normal subgroups intersecting nontrivially in the profinite residual. Using existentially closed groups, it shows existence of a non-sofic group implies a countable existentially closed non-sofic group whose centralizers form a dense chain of non-sofic subgroups of order type (ℚ,≤), that the class of ω-non-sofic groups is nonempty, and that there exist non-sofic groups of unbounded exponent.

Significance. If the conditional results hold, they impose strong structural constraints on any minimal non-sofic groups and establish that the class of non-sofic groups (if nonempty) is closed under certain constructions and contains examples with dense subgroup lattices and unbounded exponent. The explicit use of residual finiteness, maximality of normal subgroups, and existentially closed groups to derive these implications is a methodological strength, providing concrete tests and constructions that could inform searches for explicit non-sofic examples or further classification results in geometric group theory.

minor comments (2)

The abstract and introduction would benefit from a brief explicit statement of the precise definition of 'minimal non-sofic group' and 'ω-non-sofic group' (beyond the implicit usage in the theorems) to improve readability for readers outside the immediate subfield.
In the statement concerning the existentially closed non-sofic group, clarify whether the dense chain property holds for all centralizers or only for a specific family; the current phrasing leaves this slightly ambiguous.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments were provided in the report, so we have no point-by-point responses to address. We will incorporate any minor editorial or typographical suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; all claims conditional on external existence assumption

full rationale

The paper's core theorems (structural restrictions on minimal non-sofic groups, ω-non-sofic properties of locally graded groups, and existence implications for existentially closed examples) are explicitly framed as implications from the external hypothesis that at least one non-sofic group exists. No derivation reduces a claimed prediction or first-principles result to its own inputs by construction, fitted parameters, or self-citation chains. The maximality and residual finiteness arguments rely on standard group-theoretic facts without redefining terms or smuggling ansatzes via prior self-work. Self-citations, if present, are not load-bearing for the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The work rests on the standard axioms of group theory plus the external hypothesis that non-sofic groups exist. No numerical parameters are fitted. The new classes are definitional rather than postulated entities with independent evidence.

axioms (1)

standard math Standard axioms of group theory (associativity, identity, inverses)
All statements are made inside the category of groups.

invented entities (2)

minimal non-sofic group no independent evidence
purpose: To isolate the smallest possible non-sofic groups for structural study
Newly defined class whose properties are derived conditionally on existence.
ω-non-sofic group no independent evidence
purpose: To capture non-sofic groups possessing infinite descending normal chains with nontrivial profinite residual intersection
Newly introduced notion used to classify locally graded examples.

pith-pipeline@v0.9.0 · 5552 in / 1451 out tokens · 74781 ms · 2026-05-10T01:29:30.516521+00:00 · methodology

discussion (0)

Reference graph

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