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Stability, approximable quotients, and higher property (T)

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Pith's one-line read For every countable group there is a finitely presented overgroup with property (T) that is Frobenius stable yet has nonzero second cohomology against every unitary representation, and the same mechanism shows property (T^2) does not pass t

desk verdict The main construction is significant and the argument is sound, but it depends on a deferred lemma from the author's own preprint, which is the one thing to check. read the letter →

arxiv 2512.09180 v2 pith:CLN5PYN2 submitted 2025-12-09 math.GR

Francesco Fournier-Facio This is my paper · ORCID
classification math.GR MSC 20F6520F6720J06
keywords Frobeniusstabilityapproximablequotientsproperty(T)higher(T^2)relativelyhyperbolicgroupsCohen–LyndontripleDehnfillingsmallcancellation
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

The paper's central result is a construction: any countable group can be embedded into a finitely presented group Γ that has property (T) and is Frobenius stable, yet is as far as possible from the higher version of property (T). Specifically, for every unitary representation V of Γ, the second cohomology H^2(Γ;V) is nonzero, whereas property (T^2) would require all these to vanish. The groups are Frobenius stable for a potentially vacuous reason: they admit no nontrivial Frobenius-approximable quotients at all. The same method proves that property (T^2) does not pass to quotients, even for hyperbolic groups.

What carries the argument

The central objects are (1) the notion of an approximable quotient: a quotient that is the image of an asymptotic homomorphism to finite unitary groups with the Frobenius norm that stays away from the identity on nontrivial elements; a group with no such quotients is automatically Frobenius stable. (2) The Cohen–Lyndon triple (Γ, Λ, Λ), where the normal closure of an infinite cyclic subgroup Λ is a free product of conjugates; the excision isomorphism for such triples yields the exact sequence H^1(Γ;V) → H^1(Λ;V) → H^2(Γ̄;V), so property (T) (which forces H^1(Γ;V)=0) combined with Λ ≅ Z (which gives H^1(Λ;V) ≅ V) injects every unitary representation into the second cohomology of the Dehn-fill

What would settle it

For the quotient Γ̄ produced in Proposition C, compute H^2(Γ̄; V) for a single nontrivial unitary representation V. The theorem says it is always nonzero; a single zero would refute the central claim. Alternatively, exhibit a nontrivial Frobenius-approximable quotient of a group built by Proposition B, which would destroy the claimed (vacuous) stability.

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Extended reading notes

Core claim

The paper establishes Theorem A: every countable (recursively presented) group embeds into a (finitely presented) group Γ that has property (T), is Frobenius stable, and does not have property (T^2); in fact H^2(Γ;V) ≠ 0 for every unitary Γ-representation V. The proof splits into two independent constructions. First, using small-cancellation quotients of relatively hyperbolic groups, the author forces a simple non-approximable subgroup to normally generate the entire group, which kills all nontrivial approximable quotients and thereby makes the group vacuously Frobenius stable while preserving property (T) and relative hyperbolicity. Second, via a Cohen–Lyndon Dehn filling along an infinite

Load-bearing premise

The argument rides on the lemma that every non-elementary relatively hyperbolic group without nontrivial finite normal subgroups has an infinite cyclic subgroup whose normal closure is a free product of conjugates and whose Dehn-filled quotient is still relatively hyperbolic and injective on the peripheral subgroup; if that lemma fails, the strong non-vanishing of second cohomology does not follow.

Editorial extensions

If this is right

  • Finitely presented Frobenius stable groups with property (T) exist in profusion: the construction works over every countable group, and recursively presented groups yield finitely presented examples.
  • Property (T^2), and its stronger variant [T_2], does not pass to quotients: a hyperbolic group with [T_2] can have a hyperbolic quotient without (T^2).
  • The groups produced are not residually finite — they have no nontrivial finite quotients at all — so the known stable groups remain either virtually free, higher-rank lattice-like, or non-residually finite.
  • The non-vanishing extends to L^1 coefficients, linking the construction to fixed-point theorems for actions on low-dimensional contractible complexes.
  • Question E remains open: whether a finitely presented residually finite Frobenius stable group without property (T^2) exists; the paper suggests SL(3,Z) as a candidate.

Reading between the lines

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

  • The stability of the constructed groups is 'vacuous' in the sense that there are no nontrivial Frobenius-approximable quotients to be stable against; the real test of stability as a phenomenon lives in the residually finite setting, which the paper leaves open.
  • The Cohen–Lyndon filling step is a general mechanism: any group whose first cohomology vanishes into a class of modules, and which has a suitable peripheral cyclic subgroup, yields a quotient whose second cohomology is nonzero into that whole class. This could be exported to other relatively hyperbolic or acylindrically hyperbolic settings to obstruct higher cohomological vanishing.
  • Because property (T) itself passes to quotients while property (T^2) does not, higher cohomological vanishing is a strictly more fragile property; this suggests that any useful 'higher Kazhdan' notion may need to be formulated with quotient behaviour in mind rather than as a naive cohomological vanishing condition.
  • If the nonzero class in H^2 is the obstruction to defect diminishing, these groups would be Frobenius stable with a superlinear rate; the paper explicitly notes its non-vanishing theorem does not by itself show this, so the rate question is a natural next target.
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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

3 major / 5 minor

Summary. The paper proves Theorem A: every countable (resp. recursively presented) group embeds into a (finitely presented) group Γ with property (T) that is Frobenius stable but fails property (T²) in the strongest possible way, namely H²(Γ;V)≠0 for every unitary Γ-representation V. The construction is two-stage. Proposition B uses small cancellation over relatively hyperbolic groups to produce, from a non-Frobenius-approximable group A, a group Γ that is hyperbolic relative to a subgroup K containing a prescribed countable group C, has property (T), and has no non-trivial Frobenius-approximable quotients; such groups are automatically Frobenius stable. Proposition C then uses a cyclic Dehn filling and a Cohen–Lyndon triple to produce a quotient with non-vanishing H² for all unitary coefficients. As a by-product, Corollary D shows that property [T²] does not pass to hyperbolic quotients.

Significance. If correct, this is a substantial contribution: it provides a large class of finitely presented property (T) groups that are Frobenius stable for the vacuous reason that they admit no non-trivial approximable quotients, while being maximally far from having property (T²). It also demonstrates that higher property (T) behaves very differently from property (T) under quotients. The exposition is clear, the small-cancellation and cohomological arguments are coherent, and the paper is careful about attributing prior work and about its open questions. The main weakness is that the decisive Lemma 3.1 is imported without proof from the author's own unpublished preprint, and the abstract promises a post-scriptum that is absent from the submitted text.

major comments (3)
  1. [Section 3, Lemma 3.1] Lemma 3.1 is load-bearing for Proposition C and hence for Theorem A, but its proof is only a citation to the author's own preprint [FFS25, Lemma 3.5]. The lemma is non-obvious: it asserts the existence of an infinite cyclic Λ with (Γ,Λ,Λ) a Cohen–Lyndon triple whose Dehn filling is hyperbolic relative to the image of K and injective on K, while the standard 'suitable' subgroups in [Osi10] are not virtually cyclic. If [FFS25] is not yet available or carries extra hypotheses (e.g. torsion-free, malnormal K), Proposition C's strong non-vanishing H²(Γ̄;V)≠0 collapses. Please include a full proof, or state precisely the hypotheses and a publicly available reference with proof, or otherwise justify this step independently.
  2. [Abstract / full text] The abstract promises 'A post-scriptum relates and comments on an experience this paper had with an AI benchmark.' The submitted text contains no such post-scriptum. This is not merely cosmetic: the manuscript as submitted does not contain a section advertised by its own abstract. Please either include the promised post-scriptum or delete this sentence.
  3. [Proposition B and Remark 2.5] The finite-presentability clause 'if A has solvable word problem' relies on [BH74], which applies only to finitely generated groups. As stated, Proposition B would justify finite presentability for any A with solvable word problem, including infinitely generated ones; the proof does not cover that case. This does not affect Theorem A, since the A constructed there is a finite central extension of a finitely presented linear group, but Proposition B should be reworded to assume A is finitely generated with solvable word problem.
minor comments (5)
  1. [Introduction, §1] Typo: 'the the family of finite-dimensional unitary groups' should be 'the family of finite-dimensional unitary groups'.
  2. [Lemma 3.2] State explicitly that when V is viewed as a Γ-module, the action factors through Γ̄, so V is a trivial Λ-module. This is used in the proof of Proposition C.
  3. [Notation] Λ is used both for a hyperbolic group in Proposition 2.3 and for a normal subgroup in Proposition 2.4; consider different letters to avoid confusion in the proof of Proposition B.
  4. [Corollary D] The sentence 'Their second Betti number vanishes ... so they also have [T²]' is confusing in light of the preceding sentence already asserting [T²]; please clarify which property is being inferred from the Betti number vanishing.
  5. [References] Reference [FFS25] should indicate the current version or arXiv date, since it is cited for a key lemma that is not proved in the present paper.

Circularity Check

1 steps flagged · score 2.0 of 10

No definitional or fitted-input circularity; the main derivation is independent. The only noteworthy issue is a load-bearing lemma deferred to the author's own [FFS25].

  1. self citation load bearing [Section 3, Lemma 3.1 (used in Proposition C and Theorem A)]
    "Lemma 3.1. Let Γ be a group that is non-elementary hyperbolic relative to a subgroup K... Then there exists an infinite cyclic subgroup Λ such that (Γ,Λ,Λ) is a Cohen–Lyndon triple... Proof. This is achieved in the course of the proof of [FFS25, Theorem 3.1]: see [FFS25, Lemma 3.5]."

    Proposition C relies on Lemma 3.1 to obtain the infinite cyclic Cohen–Lyndon subgroup, and Lemma 3.2 then injects H¹(Λ;V) ≅ V into H²(Γ̄;V), forcing the strong non-vanishing. The lemma itself is not proved here; it is quoted from the author's preprint [FFS25, Lemma 3.5]. This is a load-bearing self-citation: if the lemma has an unstated hypothesis or is wrong, the main theorem's H²-nonvanishing mechanism collapses. It is not a fitted-input circularity or an equivalence-by-definition, but it is a genuine unproved dependency on the author's own prior work.

full rationale

The central derivation is not circular in the constructional sense. Stability is obtained because Proposition B produces a group with no non-trivial Frobenius-approximable quotients; the resulting stability is vacuous, but the hard part is the nontrivial construction of such a group via Lemma 2.1, Osin's small-cancellation quotient theorems, and an independent non-approximable group from [DCGLT20]. The H²-nonvanishing is likewise a genuine injection: Lemma 3.2, from [PS24], injects H¹(Λ;V) ≅ V into H² of the Dehn-filled quotient, with no fitted parameters or normalization forcing the outcome. The only real concern is Lemma 3.1, whose proof is deferred to the author's own [FFS25, Lemma 3.5]; this is a load-bearing self-citation and a transparency/verification gap, but not an equivalence-to-inputs circularity. There are no ansatz-smuggling citations, no imported uniqueness theorems used to forbid alternatives, and no renaming of a known result under new coordinates. The abstract promises a post-scriptum about an AI benchmark that is absent from the supplied text; this is a completeness issue, not circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The construction is theorem-driven: it relies on many deep geometric-group-theoretic tools but introduces no free constants, fitted data, or new postulated entities. The large axiom count reflects the heavy dependence on external and self-cited theorems rather than ad hoc postulates.

assumptions (9)
  • standard math Group cohomology exact sequence for pairs and the excision isomorphism for Cohen–Lyndon triples [PS24, Theorem A].
    Used in Lemma 3.2 to produce the injection H¹(Λ;V) ↪ H²(Γ/⟨⟨Λ⟩⟩;V), the engine of Proposition C.
  • domain assumption Osin's small-cancellation theorem over relatively hyperbolic groups [Osi10, Theorem 2.4], including suitable subgroups and preservation of finite presentability.
    Basis for Propositions 2.3 and 2.4, which eliminate approximable quotients while keeping relative hyperbolicity and property (T).
  • domain assumption Existence of an infinite cyclic Cohen–Lyndon subgroup and Dehn filling with injectivity on parabolics [FFS25, Lemma 3.5].
    Quoted from the author's own preprint; this is Lemma 3.1 here and is the load-bearing step for the H² non-vanishing.
  • domain assumption There exists a non-Frobenius-approximable group A that is a finite central extension of a finitely presented linear group [DCGLT20].
    Supplies the initial non-approximable subgroup A used in Proposition B and Theorem A; it also has solvable word problem.
  • domain assumption Boone–Higman: every group with solvable word problem embeds into a simple subgroup of a finitely presented group [BH74].
    Used to keep finite presentability when A has solvable word problem in Proposition B.
  • domain assumption Higman/HNN embedding: every countable group embeds into a finitely generated group, and every recursively presented group into a finitely presented group [Hig61, HNN49].
    Builds the subgroup H containing the prescribed countable group C.
  • domain assumption There exists a hyperbolic group with property (T) and no non-trivial finite normal subgroups, e.g. a cocompact lattice in the Cayley plane, and such lattices have [T₂] by [BS23].
    Provides the base hyperbolic Kazhdan group in Proposition B and the starting lattice in Corollary D.
  • domain assumption In a relatively hyperbolic group with no non-trivial finite normal subgroups, every non-trivial normal subgroup is suitable [CIOS23, Lemma 3.23].
    Allows Proposition 2.4 to force the simple subgroup S to normally generate the quotient.
  • standard math Property (T) passes to quotients; every non-identity element of a simple group normally generates it; non-approximability passes to supergroups; quotients of groups with no non-trivial approximable quotients also have none.
    Elementary closure facts used throughout the proof of Proposition B and Theorem A.

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Pith. "Pith review of Stability, approximable quotients, and higher property (T)." pith.science (2026). https://pith.science/paper/CLN5PYN2

@misc{pith2026251209180,
  author       = {Pith},
  title        = {Pith review of: Stability, approximable quotients, and higher property (T)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CLN5PYN2}},
  note         = {Machine review of arXiv:2512.09180}
}
abstract

We construct a wealth of groups that are finitely presented, Frobenius stable, have property (T), but are very far from having property (T$_2$). Our method also shows that property (T$_2$) does not pass to quotients. A post-scriptum relates and comments on an experience this paper had with an AI benchmark.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

17 extracted references · 5 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Arzhantseva and L

    [AP15] G. Arzhantseva and L. P˘ aunescu,Almost commuting permutations are near commuting permu- tations, J. Funct. Anal.269(2015), no. 3, 745–757. MR 3350728 [Arz14] G. Arzhantseva,Asymptotic approximations of finitely generated groups, Extended abstracts Fall 2012—automorphisms of free groups, Trends Math. Res. Perspect. CRM Barc., vol. 1, Springer, Cham...

    arXiv 2015

  2. [6]

    [CDE13] J. R. Carri´ on, M. Dadarlat, and C. Eckhardt,On groups with quasidiagonalC ˚-algebras, J. Funct. Anal.265(2013), no. 1, 135–152. MR 3049883 [CFF23] R. Coulon and F. Fournier-Facio,Infinite simple characteristic quotients, arXiv preprint arXiv:2312.11684,

    arXiv 2013

  3. [7]

    Chifan, A

    [CIOS23] I. Chifan, A. Ioana, D. Osin, and B. Sun,Small cancellation and outer automorphisms of Kazh- dan groups acting on hyperbolic spaces, arXiv preprint arXiv:2304.07455. To appear in Algebr. Geom. Topol.,

    arXiv

  4. [8]

    [CL63] D. E. Cohen and R. C. Lyndon,Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc.108(1963), 526–537. MR 170930 [CP25] M. Chapman and Y. Peled,Cocycle stability in permutations of random simplicial complexes, arXiv preprint arXiv:2509.21566,

    arXiv 1963

  5. [9]

    Dadarlat,Obstructions to matricial stability of discrete groups and almost flat K-theory, Adv

    [Dad21] M. Dadarlat,Obstructions to matricial stability of discrete groups and almost flat K-theory, Adv. Math.384(2021), Paper No. 107722,

    2021

  6. [12]

    Dogon and I

    MR 4660281 [DV25] A. Dogon and I. Vigdorovich,Connections between hyperlinearity, stability and character rigidity for higher rank lattices, arXiv preprint arXiv:2506.20843,

    arXiv

  7. [13]

    Eilers, T

    [ESSr20] S. Eilers, T. Shulman, and A. P. W. Sø rensen,C ˚-stability of discrete groups, Adv. Math.373 (2020), 107324,

    2020

  8. [15]

    Becker and A

    MR 4945117 [BL20] O. Becker and A. Lubotzky,Group stability and Property (T), J. Funct. Anal.278(2020), no. 1, 108298,

    2020

Show all 17 references
  1. [16]

    Glebe,Frobenius non-stability of nilpotent groups, Adv

    [Gle23] F. Glebe,Frobenius non-stability of nilpotent groups, Adv. Math.428(2023), Paper No. 109129,

    2023

  2. [20]

    Bader, A

    MR 4027744 [BLSW23] U. Bader, A. Lubotzky, R. Sauer, and S. Weinberger,Stability and instability of lattices in semisimple groups, J. Anal. Math.151(2023), no. 1, 1–23. MR 4682943 [BLT19] O. Becker, A. Lubotzky, and A. Thom,Stability and invariant random subgroups, Duke Math. ...

    2023 arXiv

  3. [21]

    Math.18(2025), no

    MR 4600056 [Gle25] ,A cup product obstruction to Frobenius stability, M¨ unster J. Math.18(2025), no. 1, 27–38. MR 4976172 [GMG22] M. A. Garc ´ ıa Morales and L. Glebsky,Property of defect diminishing and stability, Int. Electron. J. Algebra31(2022), 49–54. MR 4400809 [GMS19] ...

    2025

  4. [29]

    De Chiffre, L

    MR 4238917 [DCGLT20] M. De Chiffre, L. Glebsky, A. Lubotzky, and A. Thom,Stability, cohomology vanishing, and nonapproximable groups, Forum Math. Sigma8(2020), Paper No. e18,

    2020

  5. [37]

    de Cornulier, L

    MR 4080477 7 [dCGP07] Y. de Cornulier, L. Guyot, and W. Pitsch,On the isolated points in the space of groups, J. Algebra307(2007), no. 1, 254–277. MR 2278053 [DGO17] F. Dahmani, V. Guirardel, and D. Osin,Hyperbolically embedded subgroups and rotating families in groups acting ...

    2007

  6. [41]

    Fournier-Facio, M

    MR 4131398 [FFGS25] F. Fournier-Facio, M. Gerasimova, and P. Spaas,Local Hilbert-Schmidt stability, J. Algebra663 (2025), 589–629. MR 4811049 [FFS25] F. Fournier-Facio and B. Sun,Dimensions of finitely generated simple groups and their subgroups, arXiv preprint arXiv:2503.01987,

    2025 arXiv

  7. [56]

    R˘ adulescu,The von Neumann algebra of the non-residually finite Baumslag group xa, b|ab3a´1 “b 2yembeds intoR ω, Hot topics in operator theory, Theta Ser

    MR 4669335 [R˘ ad08] F. R˘ adulescu,The von Neumann algebra of the non-residually finite Baumslag group xa, b|ab3a´1 “b 2yembeds intoR ω, Hot topics in operator theory, Theta Ser. Adv. Math., vol. 9, Theta, Bucharest, 2008, pp. 173–185. MR 2436761 [Sco84] E. A. Scott,The embed...

    2008

  8. [2023]

    [BS25] ,Higher property T and below-rank phenomena of lattices, arXiv preprint arXiv:2511.20192,

  9. [2025]

    Bader, T

    [BGM12] U. Bader, T. Gelander, and N. Monod,A fixed point theorem forL 1 spaces, Invent. Math.189 (2012), no. 1, 143–148. MR 2929085 [BH74] W. W. Boone and G. Higman,An algebraic characterization of groups with soluble word problem, J. Austral. Math. Soc.18(1974), 41–53. MR 35...

    2012

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