Pith. sign in

REVIEW 2 major objections 8 minor 2 cited by

Hyperlinearity, stability and asymptotic spectral gap of higher rank lattices

T0 review · 2 major / 8 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.20843 v2 pith:TMHU4MK6 submitted 2025-06-25 math.GR math.OA

Alon Dogon , Itamar Vigdorovich This is my paper · ORCID
classification math.GRmath.OA MSC 22E4020H0522D2520E2643A3522D5546L5320C25
keywords hyperlineargroupsHilbert-Schmidtstabilitycharacterrigidityhigher-ranklatticesproperty(TFD)asymptoticrepresentationsspectralgapcentralextensions
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

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.

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 (2)
  1. [§8.3 (Proposition 8.8)] 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.
  2. [§7.1 (Theorem 1.10)] 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.
minor comments (8)
  1. [§8.1 (Corollary 8.2)] 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.
  2. [§6 (Proposition 6.6)] 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.
  3. [§1 (Case study)] 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].
  4. [§7 (Proof of Theorem 1.4)] 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.
  5. [§8 (Proof of Corollary 1.6)] 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.
  6. [§5 (Lemma 5.1)] 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.
  7. [§8.2 (Proof of Theorem 8.6)] 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)'.
  8. [Abstract] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper proves its equivalences from independent results; the only load-bearing import (Proposition 8.8) is an unproved adaptation from the first author's prior work, which is a completeness concern rather than a circular derivation.

full rationale

The derivation chain is self-contained in the sense relevant to circularity. Theorem 1.4 is not enacted by definitions: hyperfinite Hilbert–Schmidt stability (Definition 4.1) is a genuine constraint on asymptotic representations, and its equivalence with character rigidity is established through the non-commutative Schramm theorem (Theorem 3.4), the character-theoretic criterion (Theorem 4.2), and the charmenability results [BBHP22, BBH23], which are external to this paper. Theorem 1.5 likewise reduces non-hyperlinearity of the central extension to Theorem 8.6, which is proved in the paper, together with a cohomological input (Proposition 8.8) that produces non-coboundary characters from finite abelianization. Proposition 8.8 is quoted from the first author's earlier [Dog23] and its adaptation from property (T) to finite abelianization is asserted rather than written out; this is a load-bearing self-citation and an omitted proof, and it should be supplied. But it is not a circular step: the proposition neither assumes nor is equivalent to non-hyperlinearity or to Hilbert–Schmidt stability, and the remaining proof of Theorem 8.6 is given in full. No equation is shown to reduce to its own input, and no fitted parameter is renamed as a prediction. Hence the paper's central claims have independent content and the circularity score is 0.

Assumptions & free parameters 1 free parameters · 8 assumptions · 2 invented entities

This is a pure mathematics paper: no data fitting, no experimental input. The central claims rest on several deep external theorems, most importantly charmenability of higher-rank arithmetic groups [BBH23, BBHP22], property (T;FD) sources, and the Deligne non-residually finite extensions [Sto24, Rag84]. The only hand-chosen constant is θ0 in Lemma 8.1, which is harmless. New notions (hyperfinite HS stability, (T;FD)_rob) are formal definitions with independent mathematical content.

free parameters (1)
  • θ0
    A hand-picked universal constant in (0,1/2) satisfying 1−√2θ0 ≥ 1/√2 and 1−√2θ0−√(1−(1−√2θ0)^2) > 1/2, used in Lemma 8.1. Any sufficiently small value works; it is not fitted to data.
assumptions (8)
  • domain assumption Charmenability dichotomy for higher-rank arithmetic groups: every character is either von Neumann amenable or supported on the amenable radical [BBH23, Theorem B].
    Used in the proof of Theorem 1.4, Section 7, to split characters into amenable and supported-on-amenable-radical parts, then [BV22, Lemma 4.8] trivializes the radical.
  • domain assumption Property (T;FD) for the S-arithmetic lattices considered, as established in [LZ89] and [LZ03, Theorem 9.5] via the congruence subgroup property.
    The paper assumes property (T;FD) as a standing hypothesis in Theorems 1.4 and 1.5; for the concrete lattices in Corollary 1.6 it is imported from known results.
  • standard math Connes' equivalence between amenability and hyperfiniteness for von Neumann algebras [Con76].
    Used throughout, especially in Theorem 3.4, Theorem 4.2, and Lemma 7.2.
  • standard math Hadwin-Shulman character-theoretic criterion for Hilbert-Schmidt stability and the perturbation theorem [HS18a, Theorem 4], [HS18b, Theorem 1.1].
    Used in the proof of Theorem 4.2 to pass from trace approximation to actual conjugation of near representations.
  • domain assumption Stuck-Zimmer theorem on stabilizers for ergodic actions of higher-rank lattices [SZ94].
    Used in the permutational variant, Theorem 1.10 and Section 7.1, to replace [BBHP22, BBH23].
  • standard math Ozawa's algebraic characterization of property (T) [Oza16].
    Used in Theorem 6.10 to produce effective almost spectral gap for almost representations of Kazhdan groups.
  • standard math Margulis arithmeticity and Raghunathan finite generation and residual finiteness of lattices [Rag72, Mar91].
    Used in the proof of Theorem 1.4 to reduce from arbitrary irreducible lattices to arithmetic groups G(O_F).
  • domain assumption Serre's amalgam decomposition SL2(Z[1/p]) = SL2(Z) ∗_B SL2(Z) over the Iwahori subgroup B [Ser02, §II.1, Cor. 2].
    Underlies the reformulation of Hilbert-Schmidt stability of SL2(Z[1/p]) as Problem 1.7 in the case study.
invented entities (2)
  • Hyperfinite Hilbert-Schmidt stability independent evidence
    purpose: Weak stability notion that is equivalent to character rigidity for higher-rank lattices in Theorem 1.4.
    Defined in Definition 4.1 and shown equivalent to character limits via Theorem 4.2; it is a precisely defined mathematical property, not a physical entity.
  • Robust property (T;FD)_rob independent evidence
    purpose: Uniform spectral gap for almost representations, one of the equivalent conditions in Theorem 1.4.
    Defined in Definition 6.5 and characterized via ultraproducts in Proposition 6.6; its truth is checkable in examples.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hyperlinearity, stability and asymptotic spectral gap of higher rank lattices." pith.science (2026). https://pith.science/paper/TMHU4MK6

@misc{pith2026250620843,
  author       = {Pith},
  title        = {Pith review of: Hyperlinearity, stability and asymptotic spectral gap of higher rank lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TMHU4MK6}},
  note         = {Machine review of arXiv:2506.20843}
}
abstract

We prove that if the group $\mathrm{SL}_2(\mathbb Z[1/p])$ is flexibly Hilbert--Schmidt stable for some prime $p$, then it admits a non-hyperlinear finite central extension. Consequently, a positive answer to the following question would yield an explicit example of a non-hyperlinear group: If two representations of the modular group $\mathrm{SL}_2(\mathbb{Z})$ almost agree on an Iwahori subgroup $B$, must they be close to representations that agree on $B$? More generally, we investigate spectral gap properties for asymptotic representations of higher rank lattices and groups with property (T:FD). In this setting, we prove that character rigidity is equivalent to a weak form of stability.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Non-uniform higher-rank lattices are character rigid

    math.GR 2025-07 accept novelty 8.0 of 10

    Every irreducible non-uniform lattice in a higher-rank semisimple group of characteristic not 2 is character rigid.

  2. Stability, approximable quotients, and higher property (T)

    math.GR 2025-12 conditional novelty 7.0 of 10

    Every countable (recursively presented) group embeds into a finitely presented, property (T), Frobenius-stable group whose second cohomology is nonzero for every unitary representation.

Reference graph

Works this paper leans on

70 extracted references · 62 canonical work pages · cited by 2 Pith papers

  1. [1]

    Akhtiamov and A

    D. Akhtiamov and A. Dogon. On uniform Hilbert Schmidt stability of groups. Proc. Am. Math. Soc. , 150(4):1799--1809, 2022

    work page 2022

  2. [2]

    Ab \'e rt, Y

    M. Ab \'e rt, Y. Glasner, and B. Vir \'a g. Kesten's theorem for invariant random subgroups. Duke Math. J. , 163(3):465--488, 2014

    work page 2014

  3. [3]

    G. W. Anderson, A. Guionnet, and O. Zeitouni. An Introduction to Random Matrices . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2009

    work page 2009

  4. [4]

    Anantharaman and S

    C. Anantharaman and S. Popa. An introduction to \(II_1\) factors. Book draft available at https://www.math.ucla.edu/ popa/Books/IIun.pdf

    work page

  5. [5]

    Arzhantseva and L

    G. Arzhantseva and L. P a unescu. Almost commuting permutations are near commuting permutations. Journal of Functional Analysis , 269(3):745--757, 2015

    work page 2015

  6. [6]

    Argerami

    M. Argerami. Polar decomposition in a finite von neumann algebra. Mathematics Stack Exchange. URL:https://math.stackexchange.com/q/943650 (version: 2014-09-23)

    work page 2014

  7. [7]

    Bowen and P

    L. Bowen and P. Burton. Flexible stability and nonsoficity. Transactions of the American Mathematical Society , 373(6):4469--4481, 2020

    work page 2020

  8. [8]

    Bader, R

    U. Bader, R. Boutonnet, and C. Houdayer. Charmenability of higher rank arithmetic groups. Ann. Henri Lebesgue , 6:297--330, 2023

    work page 2023

Show all 70 references
  1. [9]

    Bader, R

    U. Bader, R. Boutonnet, C. Houdayer, and J. Peterson. Charmenability of arithmetic groups of product type. Invent. Math. , 229(3):929--985, 2022

    2022

  2. [10]

    Becker and M

    O. Becker and M. Chapman. Stability of approximate group actions: uniform and probabilistic. J. Eur. Math. Soc. (JEMS) , 25(9):3599--3632, 2023

    2023

  3. [11]

    Bekka and P

    B. Bekka and P. de la Harpe. Unitary representations of groups, duals, and characters , volume 250 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2020

    2020

  4. [12]

    Bekka, P

    B. Bekka, P. de la Harpe, and A. Valette. Kazhdan's property (T) , volume 11 of New Math. Monogr. Cambridge: Cambridge University Press, 2008

    2008

  5. [13]

    B. Bekka. Kazhdan's property ( T ) for the unitary group of a separable Hilbert space. Geom. Funct. Anal. , 13(3):509--520, 2003

    2003

  6. [14]

    B. Bekka. Operator-algebraic superrigidity for sl_n( z), n 3 . arXiv preprint math/0609102 , 2006

    2006 arXiv

  7. [15]

    Boutonnet and C

    R. Boutonnet and C. Houdayer. Stationary characters on lattices of semisimple lie groups. Publications math \'e matiques de l'IH \'E S , 133(1):1--46, 2021

    2021

  8. [16]

    Becker and A

    O. Becker and A. Lubotzky. Group stability and property ( T ). J. Funct. Anal. , 278(1):20, 2020. Id/No 108298

    2020

  9. [17]

    Bader, A

    U. Bader, A. Lubotzky, R. Sauer, and S. Weinberger. Stability and instability of lattices in semisimple groups. J. Anal. Math. , 151(1):1--23, 2023

    2023

  10. [18]

    Becker, A

    O. Becker, A. Lubotzky, and A. Thom. Stability and invariant random subgroups. Duke Mathematical Journal , 168(12):2207--2234, 2019

    2019

  11. [19]

    N. P. Brown and N. Ozawa. \(C^*\) -algebras and finite-dimensional approximations , volume 88 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2008

    2008

  12. [20]

    N. P. Brown. Invariant means and finite representation theory of \(C^*\) -algebras , volume 865 of Mem. Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2006

    2006

  13. [21]

    Bader and I

    U. Bader and I. Vigdorovich. Charmenability and stiffness of arithmetic groups. 2022

    2022

  14. [22]

    Chapman, Y

    M. Chapman, Y. Dikstein, and A. Lubotzky. Conditional Non - Soficity of p-adic Deligne Extensions : on a Theorem of Gohla and Thom . Preprint, arXiv :2410.02913 [math. CO ] (2024), 2024

    2024 arXiv

  15. [23]

    A. Connes. Classification of injective factors. Cases \(II_1\) , \(II_ \) , \(III_ \) , \( 1\) . Ann. Math. (2) , 104:73--115, 1976

    1976

  16. [24]

    Creutz and J

    D. Creutz and J. Peterson. Character rigidity for lattices and commensurators. American Journal of Mathematics , 146(3):687--711, 2024

    2024

  17. [25]

    De Chiffre, L

    M. De Chiffre, L. Glebsky, A. Lubotzky, and A. Thom. Stability, cohomology vanishing, and nonapproximable groups. Forum Math. Sigma , 8:37, 2020. Id/No e18

    2020

  18. [26]

    P. Deligne. Extensions centrales non residuellement finies de groupes arithm \'e tiques. C. R. Acad. Sci., Paris, S \'e r. A , 287:203--208, 1978

    1978

  19. [27]

    Dogon, M

    A. Dogon, M. Glasner, Y. Gofine, L. Hannany, and A. Levit. Non-uniform higher-rank lattices are character rigid. In preparation , 2025

    2025

  20. [28]

    Deroin and S

    B. Deroin and S. Hurtado. Non left-orderability of lattices in higher rank semi-simple Lie groups. Preprint, arXiv :2008.10687 [math. GR ] (2020), 2020

    2020 arXiv

  21. [29]

    de la Harpe, A

    P. de la Harpe, A. G. Robertson, and A. Valette. On the spectrum of the sum of generators for a finitely generated group. Israel J. Math. , 81(1-2):65--96, 1993

    1993

  22. [30]

    Dudko and K

    A. Dudko and K. Medynets. Finite factor representations of higman--thompson groups. Groups, Geometry, and Dynamics , 8(2):375--389, 2014

    2014

  23. [31]

    A. Dogon. Stability and approximation of groups and operator algebras, 2021. M.Sc. thesis, published by the Hebrew University of Jerusalem

    2021

  24. [32]

    A. Dogon. Flexible Hilbert - Schmidt stability versus hyperlinearity for property ( T ) groups. Math. Z. , 305(4):20, 2023. Id/No 58

    2023

  25. [33]

    G. Elek. The combinatoral cost. Enseign. Math. (2) , 53(3-4):225--235, 2007

    2007

  26. [34]

    Gelander

    T. Gelander. Things we can learn by considering random locally symmetric manifolds. arXiv preprint arXiv:2407.21208 , 2024

    2024 arXiv

  27. [35]

    Gerasimova and K

    M. Gerasimova and K. Shchepin. Stability of amalgamated free products and hnn extensions. 05 2023

    2023

  28. [36]

    Gerasimova and K

    M. Gerasimova and K. Shchepin. Virtually free groups are p - S chatten stable. Proc. Amer. Math. Soc. , 152(1):411--421, 2024

    2024

  29. [37]

    Gohla and A

    L. Gohla and A. Thom. High-dimensional expansion and soficity of groups. arXiv preprint arXiv:2403.09582 , 2024

    2024 arXiv

  30. [38]

    Houdayer

    C. Houdayer. Noncommutative ergodic theory of higher rank lattices. arXiv preprint arXiv:2110.07708 , pages 79--91, 2021

    2021 arXiv

  31. [39]

    Hadwin and T

    D. Hadwin and T. Shulman. Stability of group relations under small hilbert--schmidt perturbations. Journal of Functional Analysis , 275(4):761--792, 2018

    2018

  32. [40]

    Hadwin and T

    D. Hadwin and T. Shulman. Tracial stability for C^* -algebras. Integral Equations Operator Theory , 90(1):Paper No. 1, 35, 2018

    2018

  33. [41]

    A. Ioana. Almost commuting matrices and stability for product groups. Journal of the European Mathematical Society , 2024

    2024

  34. [42]

    Ioana, P

    A. Ioana, P. Spaas, and M. Wiersma. Cohomological obstructions to lifting properties for full C *-algebras of property ( T ) groups. Geom. Funct. Anal. , 30(5):1402--1438, 2020

    2020

  35. [43]

    Juschenko and N

    K. Juschenko and N. Monod. Cantor systems, piecewise translations and simple amenable groups. Ann. Math. (2) , 178(2):775--787, 2013

    2013

  36. [44]

    Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen. Mip*= re. Communications of the ACM , 64(11):131--138, 2021

    2021

  37. [45]

    V.F.R. Jones. Ten problems. In Mathematics: frontiers and perspectives , 2000

    2000

  38. [46]

    Lov \'a sz

    L. Lov \'a sz. Large networks and graph limits , volume 60 of Colloq. Publ., Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2012

    2012

  39. [47]

    Levit, R

    A. Levit, R. Slutsky, and I. Vigdorovich. Spectral gap and character limits in arithmetic groups. arXiv preprint arXiv:2308.05562 , 2023

    2023 arXiv

  40. [48]

    Lubotzky and R

    A. Lubotzky and R. J. Zimmer. Variants of Kazhdan 's property for subgroups of semisimple groups. Isr. J. Math. , 66(1-3):289--299, 1989

    1989

  41. [49]

    Lubotzky and A

    A. Lubotzky and A. Zuk. On property ( ) . 2003

    2003

  42. [50]

    G. A. Margulis. Discrete subgroups of semisimple Lie groups , volume 17. Springer Science & Business Media, 1991

    1991

  43. [51]

    D. W. Morris. Introduction to arithmetic groups . [s.l.]: Deductive Press, 2015

    2015

  44. [52]

    Manuilov and Chao Y

    V. Manuilov and Chao Y. On almost representations of property (t) groups. arXiv: Operator Algebras , 2007

    2007

  45. [53]

    Nicoara, S

    R. Nicoara, S. Popa, and R. Sasyk. On II \(_ 1 \) factors arising from 2-cocycles of \(w\) -rigid groups. J. Funct. Anal. , 242(1):230--246, 2007

    2007

  46. [54]

    Orovitz, R

    J. Orovitz, R. Slutsky, and I. Vigdorovich. The space of traces of the free group and free products of matrix algebras. Adv. Math. , 461:36, 2025. Id/No 110053

    2025

  47. [55]

    N. Ozawa. About the connes embedding conjecture---algebraic approaches---. arXiv preprint arXiv:1212.1700 , 2012

    2012 arXiv

  48. [56]

    N. Ozawa. Noncommutative real algebraic geometry of K azhdan's property ( T ). J. Inst. Math. Jussieu , 15(1):85--90, 2016

    2016

  49. [57]

    V. G. Pestov. Hyperlinear and sofic groups: a brief guide. Bulletin of Symbolic Logic , 14(4):449--480, 2008

    2008

  50. [58]

    Peterson

    J. Peterson. Lecture notes on von neumann algebras, 2013. Available at https://math.vanderbilt.edu/peters10/teaching/spring2013/vonNeumannAlgebras.pdf

    2013

  51. [59]

    Peterson

    J. Peterson. Character rigidity for lattices in higher-rank groups. preprint , 2, 2015

    2015

  52. [60]

    Peterson and A

    J. Peterson and A. Thom. Character rigidity for special linear groups. J. Reine Angew. Math. , 716:207--228, 2016

    2016

  53. [61]

    Discrete subgroups of Lie groups , volume 68

    Madabusi Santanam Raghunathan. Discrete subgroups of Lie groups , volume 68. Springer, 1972

    1972

  54. [62]

    M. S. Raghunathan. Torsion in cocompact lattices in coverings of Spin (2,n). Math. Ann. , 266:403--419, 1984

    1984

  55. [63]

    Rapinchuk

    A. Rapinchuk. On the finite dimensional unitary representations of Kazhdan groups. Proc. Am. Math. Soc. , 127(5):1557--1562, 1999

    1999

  56. [64]

    O. Schramm. Hyperfinite graph limits. Electron. Res. Announc. Math. Sci. , 15:17--23, 2008

    2008

  57. [65]

    J.-P. Serre. Trees . Springer Science & Business Media, 2002

    2002

  58. [66]

    M. Stover. Residual finiteness and discrete subgroups of lie groups. arXiv preprint arXiv:2407.07680 , 2024

    2024 arXiv

  59. [67]

    Stuck and R

    G. Stuck and R. J. Zimmer. Stabilizers for ergodic actions of higher rank semisimple groups. Ann. Math. (2) , 139(3):723--747, 1994

    1994

  60. [68]

    U ber unit \

    E. Thoma. \"U ber unit \"a re darstellungen abz \"a hlbarer, diskreter gruppen. Mathematische Annalen , 153(2):111--138, 1964

    1964

  61. [69]

    A. Thom. Examples of hyperlinear groups without factorization property. Groups Geom. Dyn. , 4(1):195--208, 2010

    2010

  62. [70]

    A. Thom. Finitary approximations of groups and their applications. In Proceedings of the I nternational C ongress of M athematicians--- R io de J aneiro 2018. V ol. III . I nvited lectures , pages 1779--1799. World Sci. Publ., Hackensack, NJ, 2018

    2018

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.