On minimal non-sofic and $\omega$-non-sofic groups

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

arxiv: 2604.19174 · v4 · pith:LYFXFRYYnew · submitted 2026-04-21 · 🧮 math.GR

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

K{\i}van\c{c} Ersoy This is my paper

Pith reviewed 2026-05-19 17:50 UTC · model grok-4.3

classification 🧮 math.GR

keywords non-sofic groupsminimal non-sofic groupsomega-non-sofic groupsresidually finite groupscentral extensionssimple groupsexistentially closed groupsprofinite residual

0 comments

The pith

If a minimal non-sofic group has a finitely generated residually finite maximal normal subgroup, then that subgroup is central and the group is a perfect central extension of a finitely generated non-amenable simple group.

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

The paper assumes non-sofic groups exist and examines their structure under this assumption. It focuses on minimal non-sofic groups, proving that any finitely generated residually finite maximal normal subgroup must be central. This forces the minimal non-sofic group to be a perfect central extension of a finitely generated non-amenable simple group. The work also establishes that locally graded non-sofic groups are omega-non-sofic, containing specific chains of normal subgroups. Finally, it shows that non-sofic groups imply the existence of certain existentially closed non-sofic groups and groups of unbounded exponent.

Core claim

Assuming the existence of non-sofic groups, 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 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 a non-sof

What carries the argument

The finitely generated residually finite maximal normal subgroup M of a minimal non-sofic group, which is proven to be central and allows reduction of G to a perfect central extension of a finitely generated non-amenable simple group.

If this is right

Locally graded non-sofic groups contain finitely generated non-sofic subgroups with strictly decreasing chains of finitely generated normal subgroups whose intersection is nontrivial and in the profinite residual.
The existence of non-sofic groups implies the existence of countable existentially closed non-sofic groups whose centralizers form a densely ordered chain of non-sofic subgroups of order type (Q, ≤).
Existence of a non-sofic group implies existence of a non-sofic group of unbounded exponent.

Where Pith is reading between the lines

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

This structure indicates that minimal non-sofic groups, if they exist, are organized around non-amenable simple groups via central extensions.
The descending chain condition in omega-non-sofic groups could serve as a practical test for detecting non-soficity in locally graded examples.
Such results might guide searches for non-sofic groups by focusing on extensions of known finitely generated non-amenable simple groups.

Load-bearing premise

Non-sofic groups exist and any minimal non-sofic group has a finitely generated residually finite maximal normal subgroup whose properties can be analyzed via residual finiteness and centrality.

What would settle it

Discovery of a minimal non-sofic group with a finitely generated residually finite maximal normal subgroup that is not central would contradict the claimed structure.

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 carves out minimal non-sofic and ω-non-sofic groups with conditional structural theorems on hypothetical non-sofic examples.

read the letter

This paper introduces minimal non-sofic groups and ω-non-sofic groups, along with some conditional structural results assuming non-sofic groups exist. The key new piece is the claim that if a minimal non-sofic group G has 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 also shows that locally graded non-sofic groups are ω-non-sofic, with specific subgroup chains, and that existence of one non-sofic group implies countable existentially closed examples with densely ordered centralizers that are non-sofic, plus examples of unbounded exponent. The paper does well by stating these clearly and tying them to standard facts about quotients, central extensions, and existentially closed groups. The derivations seem to avoid circularity and build on known preservation properties of soficity. The soft spots are that all conclusions are conditional on existence, which is still open, so the practical impact is limited to guiding searches or non-existence proofs. The extra assumption of a finitely generated residually finite maximal normal subgroup means the main structural result applies only in that case, not necessarily to all minimal non-sofic groups. Minor point: without the full proof text, one would want to verify the exact steps on how the quotient being non-amenable follows and how the centralizer chain is constructed. This is for researchers in geometric group theory and model theory who work on approximation properties of groups. A reader looking for new angles on the sofic conjecture would find some value in the definitions and implications. It deserves a serious referee to examine the proofs in detail. I recommend sending this to peer review.

Referee Report

1 major / 3 minor

Summary. The paper assumes the existence of non-sofic groups and introduces the classes of minimal non-sofic groups (non-sofic groups all of whose proper quotients are sofic) and ω-non-sofic groups. It proves that if G is a minimal non-sofic group admitting a finitely generated residually finite maximal normal subgroup M, then M is central in G and G is a perfect central extension of a finitely generated non-amenable simple group. It further shows that every locally graded non-sofic group is ω-non-sofic, in the sense that it contains a finitely generated non-sofic subgroup admitting a strictly decreasing chain of finitely generated normal subgroups whose intersection is nontrivial and contained in the profinite residual. Finally, the existence of a non-sofic group is shown to imply 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,≤), as well as the existence of a non-sofic group of unbounded exponent.

Significance. If the results hold, the paper supplies concrete structural constraints on hypothetical minimal non-sofic groups and demonstrates that the mere existence of non-sofic groups forces the existence of further examples with rich subgroup lattices and unbounded exponent. These consequences are obtained from standard facts about residual finiteness, preservation of soficity under central extensions by residually finite groups, and properties of existentially closed groups. The work therefore maps out some of the landscape that would be forced by the existence of non-sofic groups, which remains an open question in geometric group theory.

major comments (1)

§3 (minimal non-sofic groups): the proof that G/M is non-amenable (and hence that the simple quotient is non-amenable) relies on the known implication that amenable groups are sofic; this step is load-bearing for the claim that G is a central extension of a non-amenable simple group and should be accompanied by an explicit citation to the relevant theorem (e.g., Elek–Szabó or the original Gromov reference).

minor comments (3)

Introduction, paragraph 2: the definition of an ω-non-sofic group is given only after the statement of the main results; moving the definition earlier would improve readability.
§4, proof of the chain property: the intersection of the normal subgroups is asserted to lie in the profinite residual, but the precise residual (profinite versus pro-sofic) is not restated; a one-sentence clarification would remove ambiguity.
The paper would benefit from a short table or diagram summarizing the logical implications between the various classes (minimal non-sofic, ω-non-sofic, locally graded non-sofic, existentially closed non-sofic).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion regarding Section 3. We address the comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: §3 (minimal non-sofic groups): the proof that G/M is non-amenable (and hence that the simple quotient is non-amenable) relies on the known implication that amenable groups are sofic; this step is load-bearing for the claim that G is a central extension of a non-amenable simple group and should be accompanied by an explicit citation to the relevant theorem (e.g., Elek–Szabó or the original Gromov reference).

Authors: We agree with the referee that an explicit citation strengthens the argument at this load-bearing step. The proof invokes the standard fact that every amenable group is sofic. In the revised manuscript we have inserted a direct reference to this result, citing both Elek–Szabó (2006) and Gromov’s original work on the subject. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results are conditional derivations from the explicit definition of minimal non-sofic groups (non-sofic with all proper quotients sofic) combined with standard facts on maximal normal subgroups (yielding simple quotients), residual finiteness, and known preservation properties of soficity under central extensions by residually finite groups. These steps rely on external group-theoretic theorems rather than any self-definition, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the claim to the paper's own inputs. The existence assumptions are stated openly, and the conclusions about centrality and perfect central extensions of non-amenable simple groups follow directly without circular reduction. Other results on ω-non-sofic groups and existentially closed groups similarly invoke external results on existentially closed groups without internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the external assumption that non-sofic groups exist plus standard background axioms of group theory; no new free parameters or invented entities are introduced.

axioms (1)

standard math Standard axioms and definitions of groups, normal subgroups, residually finite groups, amenable groups, perfect groups, and sofic groups from prior literature
Invoked throughout the abstract to define the new classes and derive their properties

pith-pipeline@v0.9.0 · 5783 in / 1211 out tokens · 52619 ms · 2026-05-19T17:50:49.325347+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

locally graded non-sofic groups are necessarily ω-non-sofic... strictly decreasing chains of finitely generated normal subgroups whose intersection is nontrivial and lies in the profinite residual

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.