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A torsion-free non-sofic group

T0 review · 2 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A finitely presented torsion-free non-sofic group exists, constructed through property (T) and small cancellation, assuming a recently announced soficity-to-LEF criterion.

desk verdict Conditional on the unproven OpenAI criterion, this is a clean construction of a f.p. torsion-free non-sofic group—worth refereeing, but the dependence on the redacted external proposition is real and cannot be waved off. read the letter →

arxiv 2608.02025 v1 pith:GG7KVQMP submitted 2026-08-03 math.GR

Francesco Fournier-Facio This is my paper · ORCID
classification math.GR MSC 20F6520F6720F0620E26
keywords non-soficgroupssoficityconjecturetorsion-freeproperty(T)LEFfinitelypresentedsmallcancellationtheoryacylindricallyhyperbolic
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

This paper establishes that there exists a finitely presented torsion-free group that is not sofic. The construction avoids the Leavitt-algebra framework of the recent non-sofic example and instead builds the group from a universal torsion-free group, a simple torsion-free group, and a property (T) group, using small cancellation. The argument hinges on a technical criterion (Proposition 1.2) quoted from a recent announcement: under certain commutation conditions, soficity of a property (T) group forces a certain subgroup to be LEF. Since the constructed simple subgroup cannot be LEF, the group must be non-sofic. The result strengthens the prior counterexample by removing torsion, and it shows the soficity question has counterexamples even among torsion-free finitely presented groups.

What carries the argument

The load-bearing mechanism is Proposition 1.2, a soficity-to-LEF criterion (cited from a recent announcement): if Γ ≤ G are infinite property (T) groups, G is generated by Γ and elements tᵢ with tᵢΓtᵢ⁻¹ ≤ Γ, and there exists a finitely generated J ≤ G with [Γ,J] = Γ∩J = 1 and t₁Jt₁⁻¹ ≤ Γ, then soficity of G forces J to be LEF. The paper's contribution is a torsion-free instance of these hypotheses: it uses the universal finitely presented torsion-free group to embed a simple torsion-free group S into a property (T) group P, then a double HNN extension with Bass–Serre tree provides the commuting conjugate in an acylindrically hyperbolic group, and a small-cancellation quotient supplies proper

What would settle it

Construct a sofic approximation for the group G defined in Section 2, or produce any sofic group satisfying the hypotheses of Proposition 1.2 whose subgroup J is a finitely presented infinite simple group.

Watch

Extended reading notes

Core claim

The paper proves Theorem 1.3: there exists a finitely presented torsion-free non-sofic group G. The construction starts with a universal finitely presented torsion-free group U and a finitely presented simple torsion-free group S, embeds U into a property (T) group P, and forms a double HNN extension E in which two copies of P are attached so that a conjugate of S commutes with P and intersects it trivially. A small-cancellation quotient G of E is then taken that preserves property (T), remains torsion-free, and embeds S injectively. Defining Γ = π(P) and J = t₁⁻¹π(S)t₁, the group G satisfies the hypotheses of Proposition 1.2, so if G were sofic, J would be LEF; but J is a finitely presented

Load-bearing premise

The whole argument rests on Proposition 1.2, a soficity-to-LEF criterion that this paper quotes from a recent announcement and does not prove; if that criterion is false, the theorem is unproven.

Editorial extensions

If this is right

  • If Theorem 1.3 is correct, the soficity question has a negative answer among finitely presented torsion-free groups, not just groups with torsion.
  • The construction demonstrates that Proposition 1.2 is a flexible tool: it applies beyond Leavitt algebras to a small-cancellation setting, yielding new freedom in choosing the non-sofic group's algebraic properties.
  • Because the example is finitely presented, it is also a limit of marked groups, so the usual open-property argument for passing from a non-sofic group to a finitely presented one is not needed here.
  • The argument rules out soficity for the specific group G, and thus any future positive result on soficity of torsion-free groups would have to exclude this example.

Reading between the lines

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

  • The same template could be used to seek non-sofic groups with other properties that are not open in the space of marked groups, such as left-orderability or unique product, by choosing different simple torsion-free input groups—though the small-cancellation quotient may not preserve those properties automatically.
  • A fully self-contained proof of Proposition 1.2 in the literature would remove the dependence on the recent announcement and its redacted passage; the present paper shows only that the theorem follows from that criterion.
  • If Proposition 1.2 is false, Theorem 1.3 collapses, since the rest of the paper's construction is standard; the paper's exposition effectively isolates the entire risk in that one unproven criterion.
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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 / 6 minor

Summary. The paper announces the existence of a finitely presented torsion-free non-sofic group, conditional on an external soficity criterion. The construction begins with a universal finitely presented torsion-free group U, embeds it into a torsion-free property (T) group P by small cancellation, and then forms a double HNN extension E with stable letters conjugating P to two copies P_1, P_2 inside P, while also containing a finitely presented simple torsion-free subgroup S. A further Hull-type small-cancellation quotient yields a property (T) group G in which S survives injectively. Setting Γ = π(P), t_i = π(u_i), and J = t_1^{-1}π(S)t_1, the author verifies the commutation, intersection, and inclusion conditions of Proposition 1.2. If Proposition 1.2 holds, soficity of G would force J to be LEF, contradicting the fact that J ≅ S is an infinite finitely presented simple group. Hence Theorem 1.3.

Significance. Conditional on Proposition 1.2, this would be a genuine strengthening of the announced existence of a non-sofic group: the example is both finitely presented and torsion-free, and the construction avoids Leavitt algebras entirely, showing the flexibility of the underlying criterion. The internal group-theoretic work is coherent and elegant: the Γ∩J computation is correct, and the use of acylindrical hyperbolicity and small cancellation is standard. The paper also makes a fair point that non-soficity is open in the space of marked groups while torsion-freeness is not, so the direct construction is necessary. However, the paper is explicit that it relies on the same technical criterion as the OpenAI announcement, and that criterion is not proven here; as it stands, the central claim is conditional rather than established.

major comments (2)
  1. [Section 1, Proposition 1.2 and Theorem 1.3] The entire theorem rests on Proposition 1.2, which is quoted from [Ope26, Proposition 2.3], a non-peer-reviewed website announcement. The paper neither proves this proposition nor shows it follows from the cited works [Kun16, KT19]. Footnote 2 even notes that the OpenAI announcement has issued a redaction. Since Proposition 1.2 is the only bridge from the explicit construction to non-soficity, this is a load-bearing gap, not a presentation issue. A revision must include a complete proof of Proposition 1.2, or a precise derivation from published peer-reviewed results, together with the exact hypotheses. Otherwise Theorem 1.3 should be stated only as conditional on [Ope26].
  2. [Section 2, application of Proposition 1.2] The verification of the hypotheses of Proposition 1.2 is mostly sound: Γ and G are infinite property (T) groups, t_iΓt_i^{-1} ≤ Γ, and the computation of [Γ,J]=1 and Γ∩J=1 is correct. However, because Proposition 1.2 is not established in the manuscript, the application inherits any hidden or misstated hypothesis in the external announcement. For instance, the author notes that [Ope26] insists Γ and G be finitely generated and that property (T) supplies this, but the possibility of other unstated hypotheses (e.g., finite presentability of G or J) cannot be excluded without a self-contained statement and proof. This reinforces the need for an independent proof of Proposition 1.2.
minor comments (6)
  1. [Footnote 3] The sentence 'all property (T) groups are [Kaž67]' is incomplete; it should read 'all property (T) groups are finitely generated'.
  2. [Section 2] The phrase 'Since P_1 and P_2 are disjoint edge groups' would be clearer if it explicitly said P_1∩P_2={1} in their identification inside P.
  3. [Section 2] The assertion that 'By universality, P contains a subgroup of the form P_1×P_2×S' deserves one explanatory sentence: P contains a copy of U, and U contains every finitely presented torsion-free group, including P×P×S.
  4. [General] The manuscript contains no equation numbers; numbering the main displayed statements would make refereeing and citation easier.
  5. [Introduction] The historical and speculative remarks about the OpenAI announcement, especially the statement about what 'experts aware of [KT19]' would have proved, are informal and could be removed or significantly softened in a journal version.
  6. [References] References [BSr16], [HO17], and [Par11] are mentioned only as background on the Leavitt algebra connection and are not used in the construction; consider cutting them or making the connection explicit.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity: Theorem 1.3 is derived from an external soficity-to-LEF criterion (Proposition 1.2) plus an explicit group-theoretic construction; the only self-citation [FF25] is a minor supporting lemma also credited to Osin [Osi10].

full rationale

The derivation chain is not circular. Theorem 1.3 is obtained by applying the external criterion Proposition 1.2 ([Ope26, Proposition 2.3]) to an explicitly constructed group G. Nothing in the construction assumes that G is non-sofic or that J is non-LEF; the contradiction is supplied by the criterion. The internal steps are standard external results: Higman's universal finitely presented torsion-free group, Burger–Mozes/Hyde–Lodha finitely presented simple torsion-free groups, a torsion-free hyperbolic property (T) group (random groups), Osin's small-cancellation embedding, Hull's common quotient theorem, and Minasyan–Osin's acylindrical hyperbolicity. The proof that Γ ∩ J = 1 is a direct subgroup argument and does not feed the target conclusion back into the hypotheses. The only self-citation is [FF25, Proposition 2.3] for the precise statement that the universal group U embeds into a finitely presented torsion-free property (T) group P; the paper explicitly credits the method to Osin [Osi10, Theorem 2.4.5]. This is a minor supporting citation, not the central claim, and it is not equivalent to Theorem 1.3. The genuinely soft point is that Proposition 1.2 is quoted from an external OpenAI announcement rather than proved or derived from [Kun16, KT19] in this paper; the paper itself notes a redaction in that announcement (footnote 2). That is an unproven-premise/correctness risk, not a circularity, because the criterion is not the target result and is not defined in terms of it. Accordingly the circularity score is 2 (one minor, non-load-bearing self-citation), not higher.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The proof is a composition of deep external theorems. The most critical is Proposition 1.2 from [Ope26], which is unverified and from a non-standard source. The other assumptions are published results, though several are recent. No free parameters or invented entities appear.

assumptions (8)
  • domain assumption Existence of universal finitely presented torsion-free group U
    Used in Section 2 to embed all finitely presented torsion-free groups; cited to [Hig61, BS08, Chi14].
  • domain assumption Existence of finitely presented simple torsion-free group S
    Provides the non-LEF witness; cited to [BM97, HL25].
  • domain assumption Existence of torsion-free hyperbolic property (T) group H
    Random groups at density in (1/3,1/2); cited to [Zuk03].
  • domain assumption U embeds in finitely presented torsion-free property (T) group P
    Small cancellation over relatively hyperbolic groups; cited to [Osi10] and [FF25].
  • domain assumption Hull's common quotient theorem
    Produces finitely presented torsion-free quotient G of E*H with property (T) and injective on finite set; cited to [Hul16, Corollary 7.4 and Theorem 7.1(e)].
  • domain assumption Acylindrical hyperbolicity of the HNN extension E
    Used for small cancellation step; cited to [MO15].
  • domain assumption Proposition 1.2 from [Ope26]
    The main technical criterion linking soficity of G to LEF of J; unproven in this paper, cited to OpenAI announcement.
  • domain assumption Finitely presented LEF groups are residually finite
    Used to rule out J being LEF; cited to [CSC23, Proposition 7.3.8].

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Cite this review

Pith. "Pith review of A torsion-free non-sofic group." pith.science (2026). https://pith.science/paper/GG7KVQMP

@misc{pith2026260802025,
  author       = {Pith},
  title        = {Pith review of: A torsion-free non-sofic group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GG7KVQMP}},
  note         = {Machine review of arXiv:2608.02025}
}
read the original abstract

OpenAI announced the existence of a non-sofic group: the unit group of the binary Leavitt algebra. We exhibit a different source of examples (relying on the same technical criterion) which includes torsion-free groups.

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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. Nonsofic wreath products of residually finite groups

    math.GR 2026-08 conditional novelty 7.0 of 10

    For any infranormal, nonnormal Kazhdan subgroup Gamma of a Kazhdan group G, the generalized wreath product over G/Gamma is nonsofic, with explicit residually finite examples.

Reference graph

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