Stability for product groups and property $(\tau)$

Adrian Ioana

Stability for product groups and property $(\tau)$

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that Σ×Λ is not very flexibly P-stable whenever Σ has a non-abelian free quotient and Λ lacks property (τ).

desk verdict A genuine and important result—P-stability is not closed under direct products—proved with clean, modular arguments and honest limitations; it deserves serious refereeing and likely acceptance after minor revision. read the letter →

arxiv 1909.00282 v1 pith:UUGCKZM3 submitted 2019-08-31 math.GR math.OA

Adrian Ioana This is my paper · ORCID

classification math.GRmath.OA MSC 20B3022D55

keywords P-stabilitypermutationstabilityproperty(tau)directproductsasymptotichomomorphismssymmetricgroupsexpandergraphsflexible

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 establishes that permutation stability, the property that approximate homomorphisms to finite symmetric groups are always close to genuine homomorphisms, is not preserved by direct products. Its main theorem says that if Σ is a finitely generated group with a non-abelian free quotient and Λ is a finitely generated group without property (τ), then Σ × Λ is not very flexibly P-stable. In particular, since free groups and abelian groups are P-stable, the groups F_m × Z^d and F_m × F_n are not P-stable, answering an open question about direct products in the negative. The same mechanism gives the first non-amenable, residually finite groups that fail flexible P-stability, and it also settles the remaining Baumslag-Solitar cases and covers braid groups.

What carries the argument

The load-bearing objects are two. The first is the rigidity theorem for asymptotic homomorphisms of (Γ ⋆ Z) × Λ into Sym(X_n), where X_n = Γ/Γ_n and Γ has property (τ) with respect to {Γ_n}: if the asymptotic homomorphism is close on X_n to the restriction of a homomorphism on larger sets, then it is close to a homomorphism that extends the original action. The second is the construction of a twist permutation ρ_n from an almost-invariant subset of X_n of density between 1/7 and 1/6, which exists whenever Λ lacks property (τ) with respect to quotients given by homomorphisms q_n : Λ → X_n. The expander result of [Ka05], that symmetric groups carry bounded-degree Cayley graphs forming an expander family, supplies the property-(τ) quotients that make the general case go through, while the special case of infinite cyclic quotients uses the Selberg property for subgroups of SL_2(Z).

What would settle it

Compute, for the asymptotic homomorphism σ_n constructed in Lemma 6.1 for F_3 × Z, the quantity max_{h ∈ Z} d_H(σ_n(t,e)σ_n(e,h), σ_n(e,h)σ_n(t,e)); the proof shows it is at least 1/126 infinitely often. If one could nevertheless find homomorphisms τ_n on larger sets with d_H(σ_n(g), τ_n(g)|_{X_n}) → 0 for every generator g, the rigidity theorem would fail and Theorem A would be open.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem A: for finitely generated groups Σ and Λ, if Σ admits a non-abelian free quotient and Λ does not have property (τ), then Σ × Λ is not very flexibly P-stable. The proof proceeds by fabricating asymptotic homomorphisms from (F_L ⋆ Z) × Λ to finite symmetric groups that act on coset spaces X_n = Γ/Γ_n by left-right multiplication, and then twisting one generator by a permutation ρ_n built from an almost-invariant subset of X_n. A rigidity theorem shows that any homomorphism on a larger set that approximates the twisted asymptotic homomorphism would have to be close to a homomorphism extending the original action; but the twist is constructed to remain far from every such homomorphism, with commutator distance at least 1/126 infinitely often. The consequence is that P-stability is not closed under direct products, since F_m and Z^d, and F_m and F_n, are individually P-stable while their products are not.

Load-bearing premise

The load-bearing premise is the expander result of [Ka05], that symmetric groups Sym(n) admit bounded-degree Cayley graphs forming an expander family; if that failed, the proof would only reach products whose second factor has an infinite cyclic quotient.

Editorial extensions

If this is right

If Theorem A is correct, then F_m × Z^d and F_m × F_n are not P-stable for m,n ≥ 2 and d ≥ 1, even though each factor is P-stable.
P-stability is therefore not closed under direct products, settling the open question about direct products in the negative.
The groups covered are the first non-amenable, residually finite groups known not to be flexibly P-stable.
The Baumslag-Solitar groups BS(m,n) with |m| = |n| ≥ 2 are not P-stable, completing the classification of P-stability for these groups.
The braid groups B_n and the pure braid groups PB_n are not very flexibly P-stable for every n ≥ 3.

Reading between the lines

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

The proof suggests that the obstruction is the combination of a large factor, one with a non-abelian free quotient, and a factor possessing almost-invariant finite quotients; whether products of two amenable P-stable groups behave differently remains open, and the paper's methods do not reach it.
The explicit construction of asymptotic homomorphisms from products to symmetric groups can probably be transplanted to other finite quotient families with expansion, such as linear groups over shrinking moduli, to produce further non-P-stable products.
A concrete testable extension is whether F_m × F_n is also Hilbert-Schmidt unstable; the paper notes this as likely and cites supporting evidence from stability in tracial von Neumann algebras.
The commuting-subgroup phenomenon of Corollary C can be read as a finitary lifting obstruction, with the twist permutation supplying the obstruction for products of large groups with non-τ factors.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know about this paper: it answers a named open question from Becker–Lubotzky–Thom by showing that P-stability is not closed under direct products, and it gives the first non-amenable residually finite groups that are not flexibly P-stable. That is a real advance, not a repackaging. The main theorem is broad: if Σ has a free non-abelian quotient and Λ lacks property (τ), then Σ×Λ is not very flexibly P-stable. The corollaries (free times abelian, free times free, Baumslag–Solitar in the remaining case, braid groups) follow cleanly.

What the paper does well is the architecture. The proof separates into two reusable pieces: a rigidity theorem (Theorem 5.1) saying that an asymptotic homomorphism close to a homomorphism on a property-(τ) factor must be close to one extending the prescribed action, and a construction lemma (Lemma 6.1) producing asymptotic homomorphisms with a nontrivial obstruction from non-τ almost-invariant sets. The constants are explicit, the lemmas are stated sharply, and the reliance on Kassabov's expander family is explicit and published. The citation pattern is appropriate: the author cites his own earlier work only as a technique, not as a load-bearing result. I traced the main dependencies and found no circularity.

Soft spots are minor. The abstract states a weaker hypothesis than Theorem A—it mentions infinite cyclic quotients, while the theorem only needs no property (τ)—which might confuse a reader but does not affect correctness. There is a small handedness issue in Lemma 6.1: Lemma 2.6 yields left-almost-invariant sets while the construction uses right-almost-invariance. The stress-test note is right that replacing C_n with C_n^{-1} fixes it, so this is cosmetic. The full generality does rest on Kassabov's theorem, but that is a standard published input, not a hidden assumption. The author also honestly notes that the amenable–amenable direct product question remains open.

The intended reader is a geometric group theorist or someone working on stability, sofic groups, or property (τ). This is exactly the kind of paper a serious journal should send to referees. My recommendation: engage with it, referee it, and accept after minor presentation fixes.

Referee Report

0 major / 4 minor

Summary. The paper studies permutation stability (P-stability) and its flexible variants for countable groups. The main result, Theorem A, states that if a finitely generated group Σ admits a non-abelian free quotient and a finitely generated group Λ does not have property (τ), then Σ×Λ is not very flexibly P-stable. The proof combines two ingredients. First, Theorem 4.1 gives a structural rigidity statement for permutations of a finite group G that almost commute with the left regular representation; Theorem 5.1 converts this into a rigidity statement for asymptotic homomorphisms of (Γ∗Z)×Λ. Second, Lemma 6.1 constructs, from a failure of property (τ) along a sequence of finite quotients of Λ, an asymptotic homomorphism of (Γ∗Z)×Λ that is far from commuting on the Z-factor. These pieces are assembled in Theorem 7.1, and Theorem A follows by applying Kassabov's expander construction for symmetric groups in the general case and the Selberg property in the special case where Λ has an infinite cyclic quotient. The paper derives Corollary B, including non-P-stability of F_m×Z^d and F_m×F_n, thereby answering a question of Becker, Lubotzky and Thom, and Theorem D, giving examples that are not weakly very flexibly P-stable.

Significance. If the result stands, it is a substantial contribution to the theory of permutation stability: it settles the direct-product question of Becker, Lubotzky and Thom in the negative and provides the first non-amenable residually finite groups that are not flexibly P-stable. The paper also gives a clean separation between P-stability and Hilbert-Schmidt stability, since F_m×Z^d is HS-stable by prior work but not P-stable. The proof strategy is modular and convincing: explicit constants are tracked through Theorem 4.1 and Lemma 6.1, the rigidity argument in Theorem 5.1 is well separated from the construction in Lemma 6.1, and the key external inputs (Kassabov's expanders, Selberg property, Bourgain-Varjú expansion) are used in a transparent way. The self-contained proofs of Lemmas 2.6, 4.2 and 5.2 add to the reliability of the paper. I found no circularity or hidden ad hoc assumptions: the central claims follow from previously established results that do not include the target theorems.

minor comments (4)

[§6, Lemma 6.1] The proof invokes Lemma 2.6 to obtain sets C_n satisfying (6.1), but Lemma 2.6 produces left-almost-invariant sets, whereas (6.1) requires right-almost-invariance. This is a genuine but local gap: it is repaired by replacing C_n with C_n^{-1}, which preserves the density bounds and condition (c). I ask the author to add this one-sentence justification.
[§7, proof of Theorem 7.1] In the restatement of condition (1) after applying Lemma 6.1, the variables are mis-stated: it should read "for all g∈Γ, h∈Λ, x∈X_n" rather than "for all g∈Σ, h∈Λ, x∈X_n", since σ_n is defined on (Γ∗Z)×Λ and p_n is defined on Γ. The subsequent use of the condition makes the intended meaning clear, but the displayed line is confusing as written.
[Abstract and Theorem A] The abstract states the main class as "Σ admits a non-abelian free quotient and Λ admits an infinite cyclic quotient", while Theorem A assumes only that Λ does not have property (τ), which is stronger. The abstract therefore understates the scope of the theorem; aligning the abstract with Theorem A would avoid confusing readers about the role of property (τ).
[§7, proof of Corollary B] There is a typo in "ﬂexbily P-stable" in the proof of part (3); it should be "flexibly P-stable".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Theorem A is derived from external expander/tau results and in-paper rigidity and construction lemmas.

full rationale

The paper's derivation chain is self-contained in the relevant sense: the main results are proved from external theorems that do not include the target conclusion, plus lemmas proved inside the paper. Theorem A is assembled from Theorem 7.1, which combines Theorem 5.1 (a rigidity statement proved from Theorem 4.1, Lemmas 4.2-4.3, and Lemma 5.2) with Lemma 6.1 (a construction of asymptotic homomorphisms from the failure of property (tau)). The only non-trivial external input in the general case is Kassabov's theorem, explicitly cited as '[Ka05, Theorem 2]', that the symmetric groups admit bounded-degree expander Cayley graphs; this supplies a group Gamma = F_L with property (tau) with respect to the kernels of the maps to Sym(n). This is an independent published result, not an assumption of the theorem's conclusion. In the special case where Lambda has an infinite cyclic quotient, the paper uses the Selberg property from '[LW93]' instead. The author's own prior work appears only as a proof-technique reference inside the self-contained proof of Lemma 4.3 ('We follow closely the proofs of [Hj03, Lemma 2.5] and [Io06, Theorem 1.3]'), and the cited result [IS19] in Remark 1.4 concerns HS-stability and is not load-bearing for Theorem A. No parameter is fitted to a subset of data and later called a prediction, and no definition is made in terms of the target non-stability. The claimed implications are genuinely derived: the asymptotic homomorphism constructed in Lemma 6.1 is shown to have self-commutation defect bounded below by 1/126, while Theorem 5.1 shows any very flexible stabilization would force that defect to tend to zero; the contradiction is a real theorem, not a tautology.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several deep, standard external theorems about expanders and property (τ). No free parameters are fitted to data, and no new entities are postulated.

assumptions (5)

standard math Kassabov's theorem [Ka05, Theorem 2]: the symmetric groups Sym(n) admit Cayley graphs with uniformly bounded degree that form an expander family.
Used in the proof of Theorem A, Section 7, to construct Γ = F_L with property (τ) relative to {ker(π_n)}.
standard math Bourgain-Varju theorem [BV10, Theorem 1]: expansion in SL_d(Z/qZ) for arbitrary q.
Used in Theorem D to show the group Γ has property (τ) relative to the congruence quotients.
standard math Selberg property for SL_2(Z): property (τ) with respect to congruence subgroups [LW93].
Used in the special case of Theorem A when Λ has an infinite cyclic quotient.
standard math Abert-Elek theorem [AE10, Theorem 4] and Lemma 2.3: existence of almost invariant sets of prescribed density for descending chains.
The proof of Lemma 2.6 generalizes this to arbitrary sequences of finite index normal subgroups.
standard math Arzhantseva-Paunescu [AP14]: abelian groups and free groups are P-stable.
Used to state that the factors in Corollary B are P-stable, so the non-stability of the product answers the closure question.

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Cite this review

Pith. "Pith review of Stability for product groups and property $(\tau)$." pith.science (2026). https://pith.science/paper/UUGCKZM3

@misc{pith2026190900282,
  author       = {Pith},
  title        = {Pith review of: Stability for product groups and property $(\tau)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UUGCKZM3}},
  note         = {Machine review of arXiv:1909.00282}
}

abstract

We study the notion of permutation stability (or P-stability) for countable groups. Our main result provides a wide class of non-amenable product groups which are not P-stable. This class includes the product group $\Sigma\times\Lambda$, whenever $\Sigma$ admits a non-abelian free quotient and $\Lambda$ admits an infinite cyclic quotient. In particular, we obtain that the groups $\mathbb F_m\times\mathbb Z^d$ and $\mathbb F_m\times\mathbb F_n$ are not P-stable, for any integers $m,n\geq 2$ and $d\geq 1$. This implies that P-stability is not closed under the direct product construction, which answers a question of Becker, Lubotzky and Thom. The proof of our main result relies on a construction of asymptotic homomorphisms from $\Sigma\times\Lambda$ to finite symmetric groups starting from sequences of finite index subgroups in $\Sigma$ and $\Lambda$ with and without property $(\tau)$. Our method is sufficiently robust to show that the groups covered are not even flexibly P-stable, thus giving the first such non-amenable residually finite examples.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Ab\' e rt and G

M. Ab\' e rt and G. Elek: Dynamical properties of profinite actions , Erg. Th. Dynam. Sys., 32 (2012), 1805-1835

work page 2012

[2] [2]

Arzhantseva and L

G. Arzhantseva and L. Paunescu: Almost commuting permutations are near commuting permutations , J. Funct. Anal., 269 (3):745-757, 2015

work page 2015

[3] [3]

Arzhantseva and L

G. Arzhantseva and L. Paunescu: Constraint metric approximations and equations in groups , J. Algebra 516 (2018), 329-351

work page 2018

[4] [4]

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O. Becker and A. Lubotzky: Group stability and property (T) , preprint arXiv:1809.00632, to appear in J. Funct. Anal

work page Pith review arXiv

[5] [5]

Stability and Invariant Random Subgroups

O. Becker, A. Lubotzky and A. Thom: Stability and invariant random subgroups , preprint arXiv:1801.08381, to appear in Duke Math. J

work page Pith review arXiv

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Flexible stability and nonsoficity

L. Bowen and P. Burton: Flexible stability and nonsoficity , preprint arXiv:1906.02172

work page Pith review arXiv 1906

[7] [7]

Abelian groups are polynomially stable

O. Becker and J. Mosheiff: Abelian groups and polynomially stable , preprint arXiv:1811.00578

work page Pith review arXiv

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Baumslag and S

B. Baumslag and S. J. Pride: Groups with two more generators than relators , J. London Math. Soc. (2) 17 (1978), no. 3, 425-426

work page 1978