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REVIEW 3 major objections 5 minor 49 references

Towards super-approximation in positive characteristic

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Pith's one-line read Under a Zariski-density and trace-field condition, the Cayley graphs of a finitely generated subgroup of $\mathrm{GL}_{n_0}(\mathbb{F}_p(t))$ modulo square-free admissible polynomials form an expander family, equivalently the spectral gap…

desk verdict Genuine new result, serious machinery, but the proof has a load-bearing gap in Lemma 60 that the stress-test correctly identifies. read the letter →

arxiv 1908.07014 v2 pith:HDWLUEUS submitted 2019-08-19 math.GR

Brian Longo , Alireza Salehi Golsefidy This is my paper · ORCID
classification math.GR MSC 22E4020G3005C81
keywords super-approximationexpandergraphspositivecharacteristicfunctionfieldsCayleyspectralgapstrongapproximationsubfield-typesubgroups
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a super-approximation theorem for finitely generated subgroups of the general linear group over a rational function field $\mathbb{F}_p(t)$ with $p>5$. Provided the Zariski closure is connected, simply connected and absolutely almost simple, and provided the traces of the adjoint representation of the group generate the whole field $\mathbb{F}_p(t)$, the congruence quotients modulo square-free polynomials with suitably restricted irreducible factors form expander graphs. In spectral terms, the random-walk operator on each quotient has a second-largest eigenvalue uniformly bounded away from 1. This is the first function-field result of this kind beyond the $\mathrm{SL}_2$ case, and it isolates what must be added to the characteristic-zero method when subfield-type subgroups appear.

What carries the argument

The load-bearing mechanism is a dichotomy for proper subgroups of the finite quotients. A subgroup is structural if it lies in the $K(\ell)$-points of a proper algebraic subgroup of bounded complexity, and subfield type if, after passing to the adjoint group, it sits between the commutator subgroup and the full points of a model over a proper subfield. The trace-field hypothesis makes strong approximation applicable, so $\pi_f(\Gamma)$ splits as a product of the $G_\ell(K(\ell))$, and this is where that hypothesis enters. Structural subgroups are escaped through invariant-theoretic descriptions: every proper positive-dimensional algebraic subgroup either fixes a line in one of finitely many irreducible representations or fixes a point in one of finitely many affine representations, the latter needed because wedge powers of the adjoint representation need not be completely reducible in positive characteristic. A ping-pong argument produces a free generating set whose random walk hits any proper algebraic subgroup with exponentially small probability. Subfield-type subgroups are kept small by the admissibility condition on degrees, and a modified multi-scale product theorem converts the escape bounds into a uniform spectral gap.

What would settle it

Take any concrete $\Gamma$ for which the trace-field condition can be verified, choose a generating set $\Omega$, and compute the second-largest eigenvalue of the averaging operator on $\mathrm{Cay}(\pi_f(\Gamma),\pi_f(\Omega))$ for square-free $f$ whose irreducible factors have pairwise distinct degrees $>1$ with no prime factor below a chosen $c_0$; if the eigenvalues tend to 1 along some sequence, the uniform spectral gap asserted by the theorem fails.

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Extended reading notes

Core claim

The main result is Theorem 1. Let $\Omega$ be a finite symmetric subset of $\mathrm{GL}_{n_0}(\mathbb{F}_p[t,1/r_0(t)])$ with $p>5$, and let $\Gamma=\langle\Omega\rangle$. If the Zariski closure of $\Gamma$ is connected, simply connected and absolutely almost simple over $\mathbb{F}_p(t)$, and if the field generated by $\mathrm{Tr}(\mathrm{Ad}(\Gamma))$ equals $\mathbb{F}_p(t)$, then there is a constant $c_0$ such that the supremum of $\lambda(P_\Omega; \prod_i \mathrm{GL}_{n_0}(\mathbb{F}_p[t]/\langle\ell_i\rangle))$ over all irreducible polynomials $\ell_i$ not dividing $r_0$, with degrees strictly increasing, larger than 1, and with no prime factor below $c_0$, is strictly less than 1. The equivalent formulation Theorem 1' states that the Cayley graphs $\mathrm{Cay}(\pi_f(\Gamma),\pi_f(\Omega))$ form an expander family as $f$ ranges over square-free $c_0$-admissible polynomials. The conclusion is explicitly about uniformity: the spectral gap does not deteriorate as the modulus grows.

Load-bearing premise

The proof depends on the assumption that the traces of the adjoint representation of $\Gamma$ generate the whole field $\mathbb{F}_p(t)$; if that fails, the congruence quotients may not have the product structure the argument needs.

Editorial extensions

If this is right

  • For every admissible modulus $f$, the Cayley graph $\mathrm{Cay}(\pi_f(\Gamma),\pi_f(\Omega))$ has a spectral gap bounded below by a constant depending only on $\Gamma$, not on $f$.
  • Choosing moduli whose irreducible factors have degrees with large prime factors yields explicit expander families of size growing like $|f|^{\dim G}$.
  • The trace-field condition, not just the Zariski closure, is what determines whether super-approximation holds in this setting; equal Zariski closures are no longer sufficient.
  • The paper's Section 4 product theorem applies to any family of pairwise non-isomorphic finite groups satisfying the given axioms, giving a standalone growth statement for direct sums of such groups.
  • The admissibility restriction is a genuine feature of the proof: it is exactly the condition that neutralizes subfield-type subgroups.

Reading between the lines

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

  • The trace-field hypothesis is probably stronger than necessary; a condition stated in terms of the profinite closure of $\Gamma$ might be the right replacement, and the paper's Question 5 points toward such a formulation.
  • If the affine representations in the invariant-theoretic step are never actually needed, as Question 29 asks, the escape argument would simplify and might allow the admissibility restriction to be relaxed.
  • The multi-scale product theorem could be tested independently of function fields on products of non-isomorphic finite simple groups, e.g. $\mathrm{PSL}_2(q_i)$ with distinct $q_i$, to see whether the growth exponent can be made effective.
  • The paper suggests that subfield-type subgroups, rather than approximate subgroups, are the main obstruction to super-approximation over fields with nontrivial subfields; a similar phenomenon should appear in other global fields.
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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 claims a super-approximation theorem over a global function field: for a finitely generated subgroup Γ of GL_{n0}(F_p[t, 1/r0(t)]), p > 5, whose Zariski closure is connected, simply-connected, absolutely almost simple, and whose adjoint trace field is F_p(t), the Cayley graphs of Γ modulo square-free polynomials whose irreducible factors have degrees with no small prime factors form a family of expanders. The proof follows the Bourgain–Gamburd–SGV12–Varjú architecture: Weisfeiler strong approximation gives the product structure of congruence quotients; a Larsen–Pink dichotomy classifies proper subgroups as structural or subfield type; new positive-characteristic results (Propositions 9, 23, 28) refine the algebraic description; a modified Varjú multi-scale theorem (Proposition 33) supplies the product theorem; and Proposition 6 gives escape from purely structural subgroups. The final Lemma 60 is meant to pass from structural subgroups to arbitrary proper subgroups using the admissibility condition on degrees.

Significance. If the main theorem were correct, it would be the first higher-rank super-approximation result over F_p(t), going beyond Bradford's SL_2 result with prime degree factors, and it would extend the SGV12/Varjú machinery to positive characteristic with new subtleties (non-complete reducibility, Weil-restriction obstructions, subfield subgroups). The paper contains genuinely useful technical contributions that appear internally sound: the field-descent proof of Proposition 9, the subfield-intersection result Proposition 23/Corollary 25, and the invariant-theoretic description of positive-dimensional subgroups via affine representations in Proposition 28. However, the central final step that carries the theorem from structural subgroups to arbitrary subgroups is not justified as written, and the proof of the main theorem therefore currently fails.

major comments (3)
  1. [§5.2, Lemma 60, Eqs. (87)–(90)] The estimate (87) applies Kesten's free-group bound directly to the pushforward measure π_{f2}[P_{Ω'}^{(2l0)}] without any control of the quotient map. Many reduced words collapse to the same element in π_{f2}(Γ), so the maximal atom of the pushforward can be much larger than the free-group atom; to justify (87) one needs injectivity of π_{f2} on the relevant ball B_{2l0}(Ω'), which would require a bound such as deg f2 ≫ l0, but the paper neither states nor proves such an inequality, and the condition l0 ≪_Ω deg f2 does not imply it. Moreover, the passage from a free-group decay e^{-c l0} to a power |π_{f1}(Γ)|^{-c1} is dimensionally unmotivated unless deg f1 and deg f2 are related; the c0-admissibility condition imposes no relation between these two degrees, and f2 can have much larger degree than f1. Consequently the bounds (88)–(90) do not yield the uniform η needed for Theorem 58, and Lemma 60, which is precisely the step that converts escape from structural subgroups into escape from arbitrary proper subgroups, is not established.
  2. [§3.4, Proposition 31] Proposition 31 is the only place where the exponential escape from proper algebraic subgroups is obtained, yet no proof is supplied: the text says 'See proof of [SGV12, Proposition 20]' and §3.4 explicitly states that the proofs of Varjú and SGV12 are not repeated. The present setting involves local fields of positive characteristic, a fresh list of representations from Proposition 30, and affine actions without fixed points; it is not automatic that the ping-pong argument from characteristic zero carries over verbatim. Since Proposition 31 feeds directly into Proposition 6 and hence into the main theorem, a self-contained proof or at least a precise reduction to the cited statement is required.
  3. [§5.1, verification of (V3)L] The verification of Varjú's condition (V3)L for the structural families is sketched rather than proved. The text defines families H_i for subgroups of fixed dimension and bounded complexity, then says that for smaller dimensions one 'allow[s] slightly larger complexity to include the connected components of the intersections of larger dimension connected proper subgroups.' No argument is given that the required index bound [H1 : H1∩H2] < L holds uniformly, that the families remain closed under conjugation with a uniform L, or that the hierarchy index j < i can be maintained. This is a load-bearing assumption for Theorem 58, and it needs a proof, not a heuristic.
minor comments (5)
  1. [Abstract and §1.1] The abstract writes π_{f(x)}(Γ) while the body consistently uses π_f(t); please make the notation uniform.
  2. [Title page] The running header 'TOW ARDS SUPER-APPROXIMATION' contains a typo: 'TOWARDS' should be a single word.
  3. [§2.2, Theorem 7 and Definition 8] Theorem 7 is stated for subgroups of G^{Che}_{0,p}(F_p), but Definition 8 refers to φ_ℓ(H) and G^{Che}_p(K(ℓ)) without defining this latter symbol; align the notation between the theorem and the definition.
  4. [§3.5, proof of Proposition 6] In the final paragraph, the expression F_p[s(t)/r(t)]/⟨ℓ⟩ is used, but the ring should be written with the same parentheses as elsewhere, e.g. F_p[s(t)/r(t)] or Fp[s/r]; the intended meaning is clear but the notation should be cleaned up.
  5. [§5, beginning] The text states that by Proposition 6 there is a symmetric subset Ω' of Γ and then, in (84), π_f(⟨Ω'⟩)=π_f(Γ). To pass from spectral gap for Ω' to Theorem 1' for Ω, the equivalence of spectral gap under change of generators should be explicitly invoked, since the Cayley graphs in Theorem 1' are defined using π_f(Ω).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral-gap conclusion is derived from proven subgroup-escape estimates and an external multi-scale product theorem, not from its own conclusion.

full rationale

The derivation chain of Theorem 1 is not circular. The admissibility constant c0 is explicitly constructed (Section 5, Lemma 60) rather than fitted to the expansion conclusion; the trace-field hypothesis is a genuine structural assumption used to invoke Weisfeiler strong approximation, not an output of the proof. The escape from structural subgroups is proved in Sections 3.1-3.5 using Larsen-Pink and invariant-theoretic characterizations (Propositions 27, 28, 30, 31), and the subfield-type contribution is bounded in Lemma 60 via Kesten's free-group bound; no inequality in the chain is equivalent by definition to the desired spectral gap. Citations to [SGV12], [Var12], [SG17], and [SG19] are to published, independently checkable results; Proposition 31 is imported from [SGV12, Prop. 20] as an external benchmark rather than a self-referential uniqueness claim, so under the stated rules it does not raise the circularity score. The potential gap in Lemma 60 concerning injectivity of the pushforward into pi_f2(Gamma), if real, is a correctness defect, not evidence that the conclusion was assumed as an input.

Assumptions & free parameters 1 free parameters · 11 assumptions · 0 invented entities

The axiomatic ledger for this pure mathematics paper consists of the external theorems the proof imports (strong approximation, Larsen-Pink, product theorems, Varju's machine, Kesten bounds, Farah's theorem) plus the stated hypotheses (trace-field condition, p > 5, simplicity and type conditions). No invented physical or mathematical entities with independent-evidence burdens appear; the new objects (affine representations rho'_j, subfield-type subgroup families H'_i) are constructions of standard type from the ambient group. The single hand-chosen quantity is the admissibility constant c0, whose value the proof does not make effective.

free parameters (1)
  • c0 (admissibility threshold) = unspecified; guaranteed to exist depending on Omega
    The theorem asserts existence of c0 such that moduli whose factor degrees have no prime factor below c0 work. The proof requires choosing c0 larger than max(deg s, deg r) for the trace field F_p(s/r) of the escaped subgroup Gamma' and larger than 2/c1 in Lemma 60.
assumptions (11)
  • standard math Weisfeiler strong approximation (Wei84, Thm 1.1): Zariski-dense subgroups surject onto congruence quotients, giving pi_f(Gamma) congruent to product_l G_l(K(l)) for large-degree irreducible l.
    Invoked in Section 2.1 to identify pi_f(Gamma) with G_f(F_q0[t]/<f>) and to obtain the product decomposition (2)-(3); load-bearing for the Cayley quotient structure.
  • standard math Larsen-Pink dichotomy (LP11, Thm 0.6): a finite subgroup of an adjoint Chevalley group over F_pbar either preserves a proper subspace in a fixed representation or lies between the commutator and the full F_q-points of an F_q-model.
    Used in Section 2.2 to define structural versus subfield-type subgroups; the whole proof strategy, including the admissibility escape mechanism, hangs on this dichotomy.
  • standard math Larsen-Pink (LP11, Thm 0.5, Prop 2.3, Prop 3.2): uniform complexity bounds and properness of algebraic envelopes of structural subgroups.
    Used in Section 2.4 to bound the complexity of the algebraic subgroup H and to show H(K(l)) is a proper subgroup for large deg l; needed for Proposition 21 and Theorem 22.
  • standard math Weisfeiler (Wei84, Cor 4.6 resp. Lemma 4.6): simplicity of g(F_q)-modules for p > 5.
    Used in Lemma 17, Lemma 18, and Proposition 23 to force invariance and containment of Lie-algebra models; the p > 5 hypothesis enters here.
  • standard math Breuillard-Green-Tao (BGT11, Cor 2.4) and Pyber-Szabo (PS16, Thm 4): product growth (V4)_delta0 for finite simple groups of Lie type of bounded rank.
    Needed in Section 5.1 to verify Varju axiom (V4)_delta0 for the factors G_l(K(l)); the product theorem is the engine of the growth step.
  • standard math Varju multi-scale machinery (Var12, Sections 3 and 5; esp. Theorem 58 and Lemma 17): product theorems and spectral gap for products of quasi-random groups from axiomatic hypotheses.
    Theorem 58 converts escape-from-subgroups into a spectral gap; Proposition 33 is a modified version proved in Section 4 following Var12, and the paper states a correction to [Var12, Corollary 14].
  • standard math Sarnak-Xue trick (SX91), Gowers quasi-randomness (Gow08), Nikolov-Pyper (NP11): spectral gap from trace bounds and low-dimensional representation bounds.
    Used in the Section 1.5 outline to reduce super-approximation to L2-flattening and escape estimates; the quasi-randomness of G_l(K(l)) supplies the c0-quasirandomness constant.
  • standard math Kesten (Kes59, Thm 3): return probabilities of symmetric random walks on free groups decay exponentially.
    Used in Section 3.5 and Section 5.2 to bound word probabilities P^l_{Omega'} and to control the subfield-part estimate in Lemma 60.
  • domain assumption EMO05 Proposition 3.2 (uniform growth argument escaping algebraic subsets) transfers to any algebraically closed field via positive-characteristic Bezout.
    The paper flags (Section 3.1) that EMO05 is written over C and asserts the proof transfers using [Sch00], [Ful98], [Da94]; the transfer is asserted in-text rather than fully proven, so it is a stated domain assumption.
  • domain assumption Trace-field hypothesis: F_p(t) equals the field generated by traces of Ad(Gamma).
    Stated hypothesis of Theorem 1; needed so Weisfeiler strong approximation yields the full product structure and so subfield-type models are controlled (see the Weil-restriction discussion in Section 1.5).
  • domain assumption p > 5 with q0 > 7, and G connected, simply-connected, absolutely almost simple; G_l fibers all of the same absolute type after enlarging r0.
    Stated hypotheses of Theorem 1 and Theorem 22; they enter via LP11 (p > 3), Wei84 Lemma 4.6 (p > 5), Proposition 9 (q1 > 9), and the constant-type reduction in Section 2.1.

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Pith. "Pith review of Towards super-approximation in positive characteristic." pith.science (2026). https://pith.science/paper/HDWLUEUS

@misc{pith2026190807014,
  author       = {Pith},
  title        = {Pith review of: Towards super-approximation in positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HDWLUEUS}},
  note         = {Machine review of arXiv:1908.07014}
}
abstract

In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is a more precise formulation of our main result. For a positive integer $c_0$, we say a square-free polynomial is $c_0$-admissible if degree of irreducible factors of $f$ are distinct integers with prime factors at least $c_0$. Suppose $\Omega$ is a finite symmetric subset of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$, where $p$ is a prime more than $5$. Let $\Gamma$ be the group generated by $\Omega$. Suppose the Zariski-closure of $\Gamma$ is connected, simply-connected, and absolutely almost simple; further assume that the field generated by the traces of ${\rm Ad}(\Gamma)$ is $\mathbb{F}_p(t)$. Then for some positive integer $c_0$ the family of Cayley graphs ${\rm Cay}(\pi_{f(x)}(\Gamma),\pi_{f(x)}(\Omega))$ as $f$ ranges in the set of $c_0$-admissible polynomials is a family of expanders, where $\pi_{f(t)}$ is the quotient map for the congruence modulo $f(t)$.

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    Borel, Linear algebraic groups, 2nd edition, Graduate Texts in Mathematics 126 Springer-Verlag, New York, 1991

    A. Borel, Linear algebraic groups, 2nd edition, Graduate Texts in Mathematics 126 Springer-Verlag, New York, 1991

    work page 1991

  2. [2]

    Bourgain, A

    J. Bourgain, A. Gamburd, Uniform expansion bounds for Cayley graphs of _2( _p) , Annals of Mathematics 167 (2008) 625--642

    work page 2008

  3. [3]

    Bourgain, A

    J. Bourgain, A. Gamburd, Expansion and random walks in _d( /p^n ) :I, Journal of European Mathematical Society 10 (2008) 987--1011

    work page 2008

  4. [4]

    Bourgain, A

    J. Bourgain, A. Gamburd, Expansion and random walks in _d( /p^n ) :II. With an appendix by J. Bourgain, Journal of European Mathematical Society 11 , no. 5., (2009) 1057--1103

    work page 2009

  5. [5]

    Bourgain, A

    J. Bourgain, A. Gamburd, P. Sarnak, Affine linear sieve, expanders, and sum-product, Inventiones Mathematicae 179 , no. 3., (2010) 559--644

    work page 2010

  6. [6]

    Bourgain, P

    J. Bourgain, P. Varj\' u , Expansion in _d( /q ) , q arbitrary, Inventiones Mathematicae 188 , no 1, (2012) 151--173

    work page 2012

  7. [7]

    Bradford, Expansion, random walks and sieve in _2( _p[t]) , Israel Journal of Mathematics 215 (2016) 559--582

    H. Bradford, Expansion, random walks and sieve in _2( _p[t]) , Israel Journal of Mathematics 215 (2016) 559--582

    work page 2016

  8. [8]

    Breuillard, Approximate subgroups and super-strong approximation, 1--50

    E. Breuillard, Approximate subgroups and super-strong approximation, 1--50. In C. Campbell, M. Quick, E. Robertson, C. Roney-Dougal, Groups St Andrews 2013, Cambridge University Press, 2015

    work page 2013

Show all 49 references
  1. [9]

    Breuillard, B

    E. Breuillard, B. Green, T. Tao, Approximate subgroups of linear groups, Geometric And Functional Analysis 21 (2011) 774--819

    2011

  2. [10]

    Breuillard, H

    E. Breuillard, H. Oh (editors), Thin groups and superstrong approximation, Mathematical Sciences Research Institute Publication 61 , Cambridge University Press, Cambridge, 2014

    2014

  3. [11]

    R. W. Carter, Simple Groups of Lie Type, London, Wiley, 1972

    1972

  4. [12]

    Conrad, O

    B. Conrad, O. Gabber, G. Prasad, Pseudo-reductive groups, 2nd edition, 26 , New Mathematical Monographs, Cambridge University Press, Cambridge, 2015

    2015

  5. [13]

    Cover, J

    T. Cover, J. Thomas, Elements of information theory (2nd ed.), Wiley-Interscience Publication, Hoboken, USA, 2006

    2006

  6. [14]

    V. I. Danilov, Algebraic varieties and schemes, Algebraic geometry, I , Encyclopaedia Math. Sci. 23 , 167--297, Springer, Berlin, 1994

    1994

  7. [15]

    Eskin, S

    A. Eskin, S. Mozes, H. Oh, On uniform exponential growth for linear groups , Inventions Mathematicae 160 (2005) 1--30

    2005

  8. [16]

    Gowers, Quasirandom Groups, Combinatorics, Probability and Computing 17 (2008) 363--387

    W.T. Gowers, Quasirandom Groups, Combinatorics, Probability and Computing 17 (2008) 363--387

    2008

  9. [17]

    Farah, Approximate homomorphisms II: group homomorphisms, Combinatorica 20 (2000), 47--60

    I. Farah, Approximate homomorphisms II: group homomorphisms, Combinatorica 20 (2000), 47--60

    2000

  10. [18]

    Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete

    W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 2 , second edition, Springer-Verlag, Berlin, 1998

    1998

  11. [19]

    Helfgott, Growth and generation in _2( /p ) , Annals of Mathematics 167 (2008) 601--623

    H. Helfgott, Growth and generation in _2( /p ) , Annals of Mathematics 167 (2008) 601--623

    2008

  12. [20]

    Helfgott, Growth in _3( /p ) , Journal of the European Mathematical Society 13 , no

    H. Helfgott, Growth in _3( /p ) , Journal of the European Mathematical Society 13 , no. 3, (2011) 761--851

    2011

  13. [21]

    Helfgott, Growth in groups: ideas and perspectives, Bulletin of American Mathematical Society 52 , no

    H. Helfgott, Growth in groups: ideas and perspectives, Bulletin of American Mathematical Society 52 , no. 3, (2015) 357--413

    2015

  14. [22]

    Hoory, N

    S. Hoory, N. Linial, A. Widgerson, Expander graphs and their application, Bulletin of American Mathematical Society 43 , no. 4, (2006) 439--561

    2006

  15. [23]

    Humphreys, Introduction to L ie algebras and representation theory, Graduate Texts in Mathematics 9 , Springer-Verlag, New York, 1978

    J. Humphreys, Introduction to L ie algebras and representation theory, Graduate Texts in Mathematics 9 , Springer-Verlag, New York, 1978

    1978

  16. [24]

    Kesten, Symmetric random walks on groups, Transactions of the American Mathematical Society 92 (1959) 336--354

    H. Kesten, Symmetric random walks on groups, Transactions of the American Mathematical Society 92 (1959) 336--354

    1959

  17. [25]

    Kowalski, An introduction to expander graphs, preprint

    E. Kowalski, An introduction to expander graphs, preprint. https://people.math.ethz.ch/ kowalski/expander-graphs.pdf

  18. [26]

    Lam, A first course in non-commutative rings, Graduate texts in mathematics 131 , Springer-Verlag, New York, 1991

    T.Y. Lam, A first course in non-commutative rings, Graduate texts in mathematics 131 , Springer-Verlag, New York, 1991

    1991

  19. [27]

    Landazuri and G.M

    V. Landazuri and G.M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, Journal of Algebra 32 (1974) 418--443

    1974

  20. [28]

    Lang and A

    S. Lang and A. Weil, Number of points of varieties in finite fields, American Journal of Mathematics 76 (1954) 819--827

    1954

  21. [29]

    Larsen, R

    M. Larsen, R. Pink, Finite subgroups of algebraic groups, Journal of the American Mathematical Society 24 , no. 4, (2011) 1105-1158

    2011

  22. [30]

    Lindenstrauss, P

    E. Lindenstrauss, P. Varj\' u , Spectral gap in the group of affine transformations over prime fields, Annales de la facult\' e des sciences de Toulouse S\' e r. 6, 25 , no. 5, (2016), 969--993

    2016

  23. [31]

    Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Birkh\" a user, Boston, 1994

    A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Birkh\" a user, Boston, 1994

    1994

  24. [32]

    Lubotzky, Expander graphs in pure and applied mathematics, Bulletin of American Mathematical Society 49 (2012) 113--162

    A. Lubotzky, Expander graphs in pure and applied mathematics, Bulletin of American Mathematical Society 49 (2012) 113--162

    2012

  25. [33]

    Matthews, L

    C. Matthews, L. Vaserstein, B. Weisfeiler, Congruence properties of Zariski-dense subgroups, I, Proceeding of London Mathematical Society, series 3, 48 , no. 3, (1984) 514--532

    1984

  26. [34]

    Nikolov and L

    N. Nikolov and L. Pyber, Product decompositions of quasirandom groups and a Jordan type theorem, Journal of the European Mathematical Society 13 , no 4, (2011) 1063--1077

    2011

  27. [35]

    M. V. Nori, On subgroups of _n( _p) , Inventiones Mathematicae 88 (1987) 257--275

    1987

  28. [36]

    Pink, Strong approximation for Zariski dense subgroups over arbitrary global fields, Commentarii Mathematici Helvetici 75 (2000) 608--643

    R. Pink, Strong approximation for Zariski dense subgroups over arbitrary global fields, Commentarii Mathematici Helvetici 75 (2000) 608--643

    2000

  29. [37]

    Pyber, E

    L. Pyber, E. Szab\' o , Growth in finite simple groups of Lie type, Journal of American Mathematical Society 29 (2016) 95--146

    2016

  30. [38]

    Salehi Golsefidy, Super-Approximation, I: -adic semisimple case, International Mathematics Research Notices 2017 , no 23, (2017) 7190-7263

    A. Salehi Golsefidy, Super-Approximation, I: -adic semisimple case, International Mathematics Research Notices 2017 , no 23, (2017) 7190-7263

    2017

  31. [39]

    A. Salehi Golsefidy, Super-Approximation, II: The p -adic case and the case of bounded powers of square-free integers, Journal of European Mathematical Society 21 , no 7, (2019) 2163--2232

    2019

  32. [40]

    Salehi Golsefidy, Sum-product phenomena: -adic case, Accepted for publication in Journal d'Analyse Math\' e matique

    A. Salehi Golsefidy, Sum-product phenomena: -adic case, Accepted for publication in Journal d'Analyse Math\' e matique

  33. [41]

    Salehi Golsefidy and P

    A. Salehi Golsefidy and P. Varj\' u , Expansions in perfect groups, Geometric And Functional Analysis 22 , no. 6, (2012) 1832--1891

    2012

  34. [42]

    Sarnak, X

    P. Sarnak, X. Xue, Bounds for multiplicities of automorphic representations , Duke Mathematical Journal 64 , no. 1, (1991) 207--227

    1991

  35. [43]

    A. Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications 77 , With an appendix by Umberto Zannier, Cambridge University Press, Cambridge, 2000

    2000

  36. [44]

    T. A. Springer, Linear algebraic groups, Second edition, Birkh\" a user, Boston, MA, 1998

    1998

  37. [45]

    Steinberg, Lecture notes on Chevalley Groups, Yale University, Lecture notes, 1961

    R. Steinberg, Lecture notes on Chevalley Groups, Yale University, Lecture notes, 1961

    1961

  38. [46]

    Tao, Expansion in finite simple groups of Lie type, Graduate Studies in Mathematics Vol

    T. Tao, Expansion in finite simple groups of Lie type, Graduate Studies in Mathematics Vol. 164 , American Mathematical Society, 2015

    2015

  39. [47]

    Tao, Product set estimates for non-commutative groups , Combinatorica 28 , no

    T. Tao, Product set estimates for non-commutative groups , Combinatorica 28 , no. 5, (2008) 547--594

    2008

  40. [48]

    Varj\' u , Expansion in _d( _K/I) , I square-free, Journal of European Mathematical Society 14 , no

    P. Varj\' u , Expansion in _d( _K/I) , I square-free, Journal of European Mathematical Society 14 , no. 1, (2012) 273--305

    2012

  41. [49]

    Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Annals of Mathematics 120 , no

    B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Annals of Mathematics 120 , no. 2, 271--315

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