{"id":"a7101668-7300-4933-8121-b1186fbf791d","arxiv_id":"2506.20843","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For higher-rank lattices, Hilbert-Schmidt stability implies non-hyperlinearity of certain central extensions, and character rigidity is equivalent to hyperfinite Hilbert-Schmidt stability.","lead":"This paper proves conditional theorems connecting the stability of group representations to the existence of non-hyperlinear groups, and shows that character rigidity for higher-rank lattices is equivalent to a newly defined weak stability property. A positive answer to an explicit question about the modular group would produce a concrete group that cannot be approximated by finite matrices.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 8.8 is load-bearing for Theorem 1.5 but is not proved in the manuscript; its adaptation from the property-(T) case is asserted in one sentence.","rationale":"Read in full, the paper's core argument is coherent: Theorem 4.2, Proposition 5.3, Theorem 8.4 and Theorem 8.6 are given detailed proofs, and the main external inputs [BBH23, Tho10, Sto24] are standard for this area. I do not see a fatal error in the conditional statement of Theorem 1.5. The most load-bearing unproved internal step is Proposition 8.8: it is quoted from [Dog23] with an 'easy to see' adaptation, yet it is exactly the mechanism that produces nontrivial cocycles from finite abelianization. Without a written proof, a reader cannot check that the absence of property (T) does not break the statement. This does not require changing the verdict: the reader's CONDITIONAL assessment is appropriate, and the requested proof would upgrade the paper. I also note Theorem 1.10 is only sketched, but it is not needed for Theorem 1.5. The reader's weakest assumption about Deligne-type extensions is a dependency on external theorems rather than an internal gap; I partially agree because the reader flagged Proposition 8.8 in the rationale, but the sharper load-bearing concern is the unproved proposition itself.","tokens_in":34025,"tokens_out":41596,"duration_ms":439803,"concrete_test":"Write out the proof of Proposition 8.8. For 1→Z→Γ̃→Γ→1 with section s and cocycle c, prove that χ∘c is a coboundary iff the pushout T-extension splits, iff χ extends to a homomorphism Γ̃→T. Hence K={χ:[χ∘c]=0} equals the image of Hom(Γ̃,T) under restriction to Z. When Γ̃^ab is finite, Hom(Γ̃,T)≅Hom(Γ̃^ab,T) is finite, so K is a finite subgroup of the connected group Ẑ; since a finite (or proper closed) subgroup has empty interior, choose χ_n∈Ẑ∖K with χ_n→e. Verify that every step uses only finite abelianization of Γ̃ and not property (T). If this verification fails, Theorem 1.5 should be weakened or the proposition proved with additional hypotheses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central conditional claim of Theorem 1.5 relies on Proposition 8.8 to convert finite abelianization of the central extension Γ̃ into a sequence of nontrivial T-valued cocycles χ_n∘α with χ_n→e. The proposition is stated without proof; the text says only that the proof of [Dog23, Prop. 7.2] adapts because property (T) is used solely to get finite abelianization. This is a genuine load-bearing step: if the characterization of the set K={χ:[χ∘c]=0} as the image of Hom(Γ̃,T)→Ẑ, or the existence of χ_n→e outside K, failed in the absence of property (T), then Theorem 8.6 could not produce the cocycles c_n and Theorem 1.5 would be unsupported. I believe the argument is correct, but it is the least-supported step in the proof of the main theorem and should be written out. The reader's concern about existence of Deligne-type extensions is a related external dependency, but it is covered by published results; the internal unproved proposition is the sharper issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies connections between Hilbert–Schmidt stability, character rigidity, and hyperlinearity for higher-rank lattices and, more generally, for groups with property (T;FD). It introduces two new notions: hyperfinite Hilbert–Schmidt stability (Definition 4.1) and a robust version of property (T;FD), denoted (T;FD)_rob (Definition 6.5). The main result for lattices, Theorem 1.4, states the equivalence of: (1) hyperfinite HS stability, (2) character rigidity, (3) property (T;FD)_rob, and (4) the assertion that every character is a pointwise limit of finite-dimensional traces. The proof goes through a noncommutative analogue of Schramm's hyperfiniteness theorem (Theorem 3.4), a character-theoretic stability criterion (Theorem 4.2), and a spectral-gap inheritance result (Proposition 6.3). The second main result, Theorem 1.5, shows that if Γ has property (T;FD) and admits a central extension 1→Z→Γ̃→Γ→1 with Γ̃ of finite abelianization, then Hilbert–Schmidt stability of Γ implies that Γ̃ is not hyperlinear. This is applied in Corollary 1.6 to S-arithmetic groups with Deligne-type central extensions, and in Corollary 1.8 to the specific case of SL_2(Z[1/p]), where a positive answer to Question 1.7 yields an explicit non-hyperlinear group. A permutational variant (Theorem 1.10) is stated and sketched in §7.1.","tokens_in":34202,"tokens_out":11375,"duration_ms":103223,"significance":"If the results are correct, they establish the first conditional bridge from Hilbert–Schmidt stability to non-hyperlinearity outside the property (T) setting, with a concrete potential counterexample linked to a stability question about SL_2(Z[1/p]). The paper contains substantial novel technical machinery: the noncommutative Schramm theorem (Theorem 3.4), the character-theoretic criterion for hyperfinite HS stability (Theorem 4.2), local rigidity results for approximate representations (Proposition 5.3), and a criterion converting non-coboundary characters into non-stability (Theorem 8.6). The proofs of Theorems 1.4, 1.5, 8.4 and 8.6 are detailed and appear to be sound up to the issues noted below. The paper honestly relies on deep external results (notably [BBH23], [BBHP22], [SZ94], [HS18a]) and attributes them clearly. The conditional nature of the main application is a strength, not a weakness, because it exactly isolates the missing piece needed for an explicit non-hyperlinear group.","major_comments":[{"comment":"Proposition 8.8 is load-bearing for Theorem 1.5 and its corollaries, but it is stated without proof. The text in the proof of Theorem 1.5 says only that the proof of [Dog23, Prop. 7.2] adapts because property (T) is used solely to get finite abelianization. This is not sufficient for a central step. In particular, the proof of Theorem 1.5 uses Proposition 8.8 to assert both the existence of χ with [χ∘α]≠0 and the existence of a sequence χ_n→e outside the subgroup K={χ:[χ∘α]=0}. Please provide a full proof. A natural argument is to show that K is the image of the restriction map Hom(Γ̃,T)→Hom(Z,T), so finite abelianization of Γ̃ makes K finite and hence a proper subgroup of the connected group Ẑ, from which the existence of χ_n→e outside K follows; but this must be written out explicitly.","section":"§8.3 (Proposition 8.8)"},{"comment":"Theorem 1.10 is stated as a numbered main theorem, but only a sketch is given. In particular, the analogue of Theorem 4.2 (stated as Theorem 7.4) is not proved, and the implication (3)⇒(2) is justified in a few sentences using property (τ) and Proposition 6.3. Since this theorem is advertised in the introduction, the paper should either provide a complete proof or explicitly demote it to a conjecture or a remark indicating that only a sketch is available. As it stands, the reader cannot verify the equivalence (1)⇔(3) without substantial additional work.","section":"§7.1 (Theorem 1.10)"}],"minor_comments":[{"comment":"In the statement of Corollary 8.2 the bound ∥η∥_op ≤ 2∥ξ∥_op/∥p∥_{2,M} is claimed, but the proof concludes ∥η∥_op ≤ 4∥ξ∥_op/∥p∥_{2,M}. Since only uniform boundedness of η_n is needed in Theorem 8.4, this discrepancy does not affect the main results, but the constant must be corrected.","section":"§8.1 (Corollary 8.2)"},{"comment":"The equivalence of (1) and (2) in Proposition 6.6 is left to the reader. This equivalence is used to link the robust spectral-gap definition with asymptotic representations, so it would be helpful to include at least a short proof or a precise reference.","section":"§6 (Proposition 6.6)"},{"comment":"The claim that Question 1.7 is equivalent to Hilbert–Schmidt stability of SL_2(Z[1/p]) is stated as 'not hard to see' and attributed to [GS23, Proposition 3]. Since Corollary 1.8 is the headline application, please spell out the reduction or provide a more precise citation to the relevant statement in [GS23].","section":"§1 (Case study)"},{"comment":"In the proof of Theorem 1.4, the passage between characters of Γ and characters of the commensurable arithmetic group G(O_F) is described as 'straightforward'. A few lines explaining how von Neumann amenability and support on the amenable radical behave under lifting and restriction would strengthen the rigor of this step.","section":"§7 (Proof of Theorem 1.4)"},{"comment":"In the proof of Corollary 1.6, the text says 'since G is F-anisotropic', but the statement of the corollary assumes the F-rank of G is at least 1, which means G is not F-anisotropic. This appears to be a typo; the intended hypothesis is likely 'F-isotropic' or something similar. Please correct it.","section":"§8 (Proof of Corollary 1.6)"},{"comment":"In the proof of Lemma 5.1, the inequality includes the symbol '≤C.S 2∥pπ(s)∥...', which seems to be a typo for an application of Cauchy–Schwarz. Please clarify the notation.","section":"§5 (Lemma 5.1)"},{"comment":"In the proof of Theorem 8.6, the phrase 'As Γ has asymptotic property (T;FD)' appears; the term 'asymptotic property (T;FD)' is not defined in the paper and should be replaced by 'property (T;FD)'.","section":"§8.2 (Proof of Theorem 8.6)"},{"comment":"The abstract refers to a 'specific congruence subgroup H' under a commensuration, while Question 1.7 and the case study use the Iwahori subgroup B. Please unify the terminology.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is a significant contribution and the conditional main results are valuable. The two major comments—the missing proof of Proposition 8.8 and the sketch-only status of Theorem 1.10—should be addressed before the paper is accepted. The factor-2/4 discrepancy in Corollary 8.2 is a minor error but should be fixed. I am willing to look at a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this is a serious, well-written paper that proves a new equivalence circle — hyperfinite HS stability, character rigidity, robust (T;FD), and finite-dimensional trace approximation — for higher-rank lattices, and derives a striking conditional: flexible HS stability of SL2(Z[1/p]) would produce a non-hyperlinear central extension. The proofs of the main theorems are detailed and hang together. I read the full text; no fatal error jumped out.\n\nWhat is genuinely new: the noncommutative Schramm theorem (3.4), the notion of hyperfinite HS stability, the robust property (T;FD), and the local rigidity theorem for asymptotically projective representations (8.4). The architecture extends Dogon's property (T) result [Dog23] to (T;FD), which is where the non-uniform lattices live, and the concrete reformulation in terms of representations of SL2(Z) almost agreeing on an Iwahori subgroup is a nice, explicit hook.\n\nThe soft spot the stress-test flags is real: Proposition 8.8 is load-bearing for Theorem 1.5 and it is not proved. The text says the proof of [Dog23, Prop. 7.2] adapts because property (T) is only used to get finite abelianization. That is a plausible claim — the character group of Z is connected, so the vanishing set cannot contain an open neighborhood — but this step converts finite abelianization into the sequence of nontrivial cocycles that Theorem 8.6 needs. If the adaptation fails, Theorem 1.5 is unsupported. This deserves a written proof, not a remark. It is not a fatal flaw, but it is a genuine gap in the manuscript.\n\nOther issues are minor by comparison: Theorem 1.10 is labelled a theorem but only sketched (the authors say so openly); Proposition 6.6 leaves one equivalence to the reader; the heavy reliance on [BBH23, BBHP22, SZ94] is standard for this area. The reader's worry about existence of Deligne-type extensions is covered by published results [Sto24, Rag84], so I would not weigh that heavily.\n\nBottom line: this is a paper for people working on stability, hyperlinearity, and character rigidity. It deserves careful refereeing. I would send it out, and ask the referee to verify Proposition 8.8 in full detail — if it holds, the conditional results stand, and the paper is an important contribution.","headline":"A serious, well-structured paper that proves a new equivalence circle for higher-rank lattices and a striking conditional route to non-hyperlinear groups, but with one load-bearing proposition (8.8) left unproved.","tokens_in":34838,"tokens_out":2809,"would_cite":true,"duration_ms":30169,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E40","20H05","22D25","20E26","43A35","22D55","46L53","20C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that Hilbert–Schmidt stability of a higher-rank lattice with property (T;FD) implies character rigidity, and that stability of the lattice rules out hyperlinearity of any central extension with kernel $\\mathbb{Z}$ and…","keywords":["hyperlinear groups","Hilbert-Schmidt stability","character rigidity","higher-rank lattices","property (T;FD)","asymptotic representations","spectral gap","central extensions"],"falsifier":"Produce a group $\\Gamma$ with property (T;FD) that is Hilbert–Schmidt stable together with a central extension $\\widetilde{\\Gamma}$ by $\\mathbb{Z}$ with finite abelianization that is hyperlinear; Theorem 1.5 declares this impossible. Concretely, one could try to show that $\\mathrm{SL}_2(\\mathbb{Z}[1/p])$ is Hilbert–Schmidt stable (equivalently, answer Question 1.7 positively) and then construct a hyperlinear realization of its Deligne-type central extension.","tokens_in":33746,"feed_emoji":"🔗","tokens_out":18173,"duration_ms":169895,"temperature":0.7,"pith_summary":"This paper connects three open problems about higher-rank lattices: character rigidity, Hilbert–Schmidt stability, and hyperlinearity. It proves that for an irreducible lattice with property (T;FD), Hilbert–Schmidt stability implies character rigidity, and that character rigidity is equivalent to a hyperfinite version of Hilbert–Schmidt stability and to a robust form of spectral gap. It then proves that if such a group has a central extension by $\\mathbb{Z}$ with finite abelianization, stability of the base group forces the extension to be non-hyperlinear. Applied to $\\mathrm{SL}_2(\\mathbb{Z}[1/p])$, a positive answer to a concrete question about representations of $\\mathrm{SL}_2(\\mathbb{Z})$ that almost agree on an Iwahori subgroup would yield an explicit non-hyperlinear group.","feed_headline":"Stable SL2(Z[1/p]) forces a non-hyperlinear group","feed_subtitle":"A positive answer to one explicit question about modular-group representations yields a concrete non-hyperlinear group.","key_machinery":"The load-bearing objects are hyperfinite Hilbert–Schmidt stability, a weakening that only asks asymptotic representations with hyperfinite generating tuples to be correctable; the non-commutative Schramm theorem, which says a sequence of matrix tuples is hyperfinite exactly when its limiting von Neumann algebra is amenable; local rigidity for almost representations, which under property (T;FD) and stability promotes closeness of two asymptotic representations to asymptotic conjugacy; and asymptotically projective representations, whose associated $2$-cocycles become coboundaries if the representations are too close to honest ones. The bridge to hyperlinearity is the twisted group von Neumann algebra attached to a central extension, whose Connes embeddability is equivalent to hyperlinearity of the extension.","core_discovery":"For an irreducible lattice $\\Gamma$ in a center-free semisimple Lie group of real rank at least two with property (T;FD), the paper establishes that four conditions coincide: hyperfinite Hilbert–Schmidt stability; character rigidity; a robust version of property (T;FD) that gives spectral gap for almost representations; and the property that every character is a pointwise limit of normalized traces of finite-dimensional representations. In particular, ordinary Hilbert–Schmidt stability implies character rigidity. Independently, for any group with property (T;FD), Hilbert–Schmidt stability rules out hyperlinearity of any central extension with kernel $\\mathbb{Z}$ and finite abelianization; via the presentation of $\\mathrm{SL}_2(\\mathbb{Z}[1/p])$ as an amalgamated free product over an Iwahori subgroup, a positive answer to a specific stability question yields an explicit non-hyperlinear group.","pith_inferences":["If the main route is realized, the non-hyperlinear group would live inside the well-studied S-arithmetic world as a central extension of a residually finite lattice, rather than in an exotic construction, making a counterexample to 'all groups are hyperlinear' accessible to arithmetic group theory.","Because hyperfinite Hilbert–Schmidt stability holds for $F_2 \\times F_2$ while ordinary stability fails, the hyperfinite notion may be the right stability-type condition to test on higher-rank lattices: weak enough to hold in interesting cases and, by Theorem 1.4, strong enough to force character rigidity.","The equivalence with robust property (T;FD) suggests a numerical route: estimate the almost spectral gap of Laplacians of asymptotic representations on finite quotients or random matrix models; a uniform positive gap would be evidence for character rigidity and hence for the stability-type conditions in Theorem 1.4.","The local rigidity result for asymptotically projective representations turns non-coboundary $2$-cocycles into obstructions to stability, suggesting that circle-valued cohomology, not only Kazhdan's property (T), controls whether almost representations can be corrected and may yield further non-hyperlinear examples among other S-arithmetic groups."],"forward_implications":["If a higher-rank lattice with property (T;FD) is Hilbert–Schmidt stable, it satisfies character rigidity, making a finitary approximation property a certificate for a conjecture previously attacked through von Neumann algebras and ergodic theory.","A positive answer to Question 1.7, the stability of the amalgam $\\mathrm{SL}_2(\\mathbb{Z}) *_{\\widetilde{B}} \\mathrm{SL}_2(\\mathbb{Z})$ over an Iwahori subgroup, yields an explicit non-hyperlinear group, namely a central extension of $\\mathrm{SL}_2(\\mathbb{Z}[1/p])$.","For any S-arithmetic lattice of the stated type, Hilbert–Schmidt stability of the lattice implies the existence of a non-hyperlinear central extension.","Theorem 1.4 makes character rigidity for higher-rank lattices equivalent to a robust spectral-gap statement about almost representations, so proving either one settles the other.","In the permutation analogue, for higher-rank lattices with property ($\\tau$), hyperfinite permutation stability is equivalent to the statement that every ergodic invariant random subgroup is either essentially free or a weak-$*$ limit of finite-index invariant random subgroups."],"supporting_citations":[{"why":"Deligne's original non-residually finite central extensions of arithmetic groups, the historical source for the extensions Theorem 1.5 requires.","marker":"[Del78]"},{"why":"Supplies the existence of non-residually finite central extensions with finite abelianization for the S-arithmetic lattices covered by Corollary 1.6.","marker":"[Sto24]"},{"why":"Provides the non-residually finite central extensions in the cocompact case, adding to the input for Proposition 8.8.","marker":"[Rag84]"},{"why":"Theorem B gives the charmenability dichotomy for higher-rank arithmetic groups that carries the proof of Theorem 1.4.","marker":"[BBH23]"},{"why":"Connes' equivalence between amenability and hyperfiniteness for von Neumann algebras underpins the non-commutative Schramm theorem and Theorem 4.2.","marker":"[Con76]"},{"why":"Hadwin and Shulman's trace criterion for stability is generalized in Theorem 4.2 to hyperfinite stability.","marker":"[HS18a]"},{"why":"Lemma 3.3 connects hyperlinearity of a central extension to Connes embeddability of twisted group von Neumann algebras, the bridge used in Theorem 1.5.","marker":"[Tho10]"},{"why":"Shows that $\\mathrm{SL}_2(\\mathbb{Z})$ is Hilbert–Schmidt stable, which makes Question 1.7 equivalent to stability of $\\mathrm{SL}_2(\\mathbb{Z}[1/p])$.","marker":"[GS24]"},{"why":"Establishes the property (T) precursor of Theorem 1.5, whose method is extended here to property (T;FD).","marker":"[Dog23]"}],"fun_headline_variants":["Stable SL2(Z[1/p]) yields non-hyperlinear central extension","Character rigidity equals weak stability in higher rank lattices","New explicit non-hyperlinear group if modular stability question holds","Stability forces spectral gap for asymptotic reps of higher rank lattices","Hilbert-Schmidt stability rules out hyperlinearity in central extensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that the targeted higher-rank lattices admit infinite central extensions with finite abelianization that are not residually finite, of the kind Deligne produced; if those extensions did not exist, the non-coboundary characters driving the contradiction with stability would not arise.","fun_headline_variants_meta":{"raw":{"variants":["Stable SL2(Z[1/p]) yields non-hyperlinear central extension","Character rigidity equals weak stability in higher rank lattices","New explicit non-hyperlinear group if modular stability question holds","Stability forces spectral gap for asymptotic reps of higher rank lattices","Hilbert-Schmidt stability rules out hyperlinearity in central extensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000243,"raw_usage":{"total_tokens":1482,"prompt_tokens":852,"completion_tokens":630,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":539}},"tokens_in":468,"tokens_out":630,"duration_ms":6594,"temperature":1.0,"reasoning_tokens":539,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:40:36.892934+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce a group $\\Gamma$ with property (T;FD) that is Hilbert–Schmidt stable together with a central extension $\\widetilde{\\Gamma}$ by $\\mathbb{Z}$ with finite abelianization that is hyperlinear; Theorem 1.5 declares this impossible. Concretely, one could try to show that $\\mathrm{SL}_2(\\mathbb{Z}[1/p])$ is Hilbert–Schmidt stable (equivalently, answer Question 1.7 positively) and then construct a hyperlinear realization of its Deligne-type central extension.","supporting_citations":[],"review_version":1}