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arxiv: 2308.09982 · v5 · submitted 2023-08-19 · 🧮 math.GR · math.CO· math.DS· math.NT

Super approximation for SL₂times SL₂ and ASL₂

Jincheng Tang , Xin Zhang This is my paper

Pith reviewed 2026-05-24 07:16 UTC · model grok-4.3

classification 🧮 math.GR math.COmath.DSmath.NT
keywords expandersCayley graphsZariski dense subgroupssuper approximationSL(2,Z)affine special linear groupspectral gap
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The pith

When S generates a Zariski-dense subgroup of SL₂(ℤ)×SL₂(ℤ) or SL₂(ℤ)⋉ℤ², the Cayley graphs on G mod q form a family of expanders.

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

The paper shows that if a finite symmetric set S generates a Zariski-dense subgroup G inside either SL₂(ℤ)×SL₂(ℤ) or the affine group SL₂(ℤ)⋉ℤ², then the sequence of Cayley graphs Cay(G mod q, S mod q) over all integers q has a uniform positive spectral gap. This uniform expansion is the definition of super-approximation for these groups. A reader cares because the result converts an algebraic condition (Zariski density) into a combinatorial statement that holds for every modulus simultaneously, without dependence on q.

Core claim

We prove that the Cayley graphs {Cay(G(mod q), S (mod q))}_{q∈ℤ} form a family of expanders whenever S is finite and symmetric and generates a Zariski-dense subgroup G of SL₂(ℤ)×SL₂(ℤ) or of SL₂(ℤ)⋉ℤ².

What carries the argument

Zariski density of the subgroup generated by S inside the ambient algebraic group over ℚ, which forces the finite quotients G mod q to be large enough and sufficiently irreducible for known expander criteria to apply uniformly.

If this is right

  • The spectral gap remains bounded away from zero independently of the modulus q.
  • The same generating set S produces expander quotients simultaneously for every prime power and every integer modulus.
  • The result applies equally to the direct product SL₂×SL₂ and to the semidirect product ASL₂.
  • Finite symmetric generating sets that are Zariski dense automatically satisfy the super-approximation property for these two families of groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same density-plus-expansion mechanism may apply to other arithmetic groups whose finite quotients admit strong approximation.
  • Uniform expansion supplies a quantitative input for sieve methods that count integer points avoiding certain residue classes modulo varying q.
  • One could test whether the proof adapts when the ambient group is replaced by a product of three or more copies of SL₂.

Load-bearing premise

The finite set S must generate a subgroup that is Zariski dense in the target algebraic group.

What would settle it

Existence of a sequence of moduli q_n such that the second-largest eigenvalue of the normalized adjacency operator on Cay(G mod q_n, S mod q_n) tends to 1.

read the original abstract

Let $S\subset \text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$ or $\text{SL}_2(\mathbb Z)\ltimes \mathbb Z^2$ be finite symmetric and assume $S$ generates a group $G$ which is a Zariski-dense subgroup $\text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$ or $\text{SL}_2(\mathbb Z)\ltimes \mathbb Z^2$. We prove that the Cayley graphs $$\{\mathcal Cay(G(\text{mod } q), S (\text{mod } q))\}_{q\in \mathbb Z}$$ form a family of expanders.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that if S is a finite symmetric generating set for a Zariski-dense subgroup G of SL_2(Z) × SL_2(Z) or of SL_2(Z) ⋉ Z², then the family of congruence Cayley graphs {Cay(G(mod q), S(mod q))}q∈Z forms an expander family (uniform spectral gap).

Significance. If the argument is correct, the result supplies super-approximation for these product and semidirect-product groups, extending the known theory from simple groups such as SL_2 to the indicated settings. Such statements are load-bearing for constructions of explicit expanders and for applications in arithmetic geometry and diophantine approximation.

minor comments (1)
  1. The abstract states the result for both SL_2×SL_2 and ASL_2 but does not indicate whether the proofs are parallel or require separate arguments; a brief outline of the logical structure would help the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting its potential significance in extending super-approximation results. The recommendation is listed as uncertain, but no specific major comments or concerns about the argument are provided in the report. We are happy to clarify any aspects of the proof if the referee has further questions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a direct existence theorem: under the hypothesis that a finitely generated group G is Zariski-dense in SL2(Z)×SL2(Z) or ASL2(Z), the congruence Cayley graphs on G(mod q) form an expander family. No equations, fitted parameters, or self-referential definitions appear in the claim. The result is a standard super-approximation statement whose proof, if valid, rests on external tools from homogeneous dynamics and representation theory rather than reducing to its own inputs by construction. Self-citations, if present, are not load-bearing for the central claim in a manner that creates circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard facts about Zariski density and finite generation in algebraic groups over Z; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Zariski density implies strong approximation properties for the groups in question
    Invoked to ensure the reductions modulo q behave well for all q.

pith-pipeline@v0.9.0 · 5664 in / 1135 out tokens · 29942 ms · 2026-05-24T07:16:08.617821+00:00 · methodology

discussion (0)

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Forward citations

Cited by 2 Pith papers

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

  1. Expansion in $\text{SL}_2(\mathbb Z/q\mathbb Z)$ and Zaremba's conjecture

    math.NT 2026-05 unverdicted novelty 8.0

    Expansion theory for SL_2(Z/qZ) is developed and used to prove Zaremba's conjecture.

  2. Expansion in $\text{SL}_2(\mathbb Z/q\mathbb Z)$ and Zaremba's conjecture

    math.NT 2026-05 unverdicted novelty 7.0

    By proving expansion in SL₂(ℤ/qℤ) and applying Shkredov's framework, the paper confirms Zaremba's conjecture.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 1 Pith paper

  1. [1]

    1, isoperimetric inequalities for graphs, and superconcentrators

    Noga Alon and Vitali D Milman. 1, isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B , 38(1):73--88, 1985

    work page 1985

  2. [2]

    Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus

    Jean Bourgain, Alex Furman, Elon Lindenstrauss, and Shahar Mozes. Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus. Journal of the American Mathematical Society , 24(1):231--280, 2011

    work page 2011

  3. [3]

    Expansion and random walks in SL _d( Z /p^n Z ) : I

    Jean Bourgain and Alex Gamburd. Expansion and random walks in SL _d( Z /p^n Z ) : I. Journal of the European Mathematical Society , 10(4):987--1011, 2008

    work page 2008

  4. [4]

    Uniform expansion bounds for C ayley graphs of SL _2( F_p)

    Jean Bourgain and Alex Gamburd. Uniform expansion bounds for C ayley graphs of SL _2( F_p) . Ann. of Math. (2) , 167(2):625--642, 2008

    work page 2008

  5. [5]

    Expansion and random walks in SL _d( Z /p^n Z ) , II

    Jean Bourgain and Alex Gamburd. Expansion and random walks in SL _d( Z /p^n Z ) , II . J. Eur. Math. Soc.(JEMS) , 11(5):1057--1103, 2009

    work page 2009

  6. [6]

    Affine linear sieve, expanders, and sum-product

    Jean Bourgain, Alex Gamburd, and Peter Sarnak. Affine linear sieve, expanders, and sum-product. Invent. Math. , 179(3):559--644, 2010

    work page 2010

  7. [7]

    The sum-product theorem in Z /q Z with q arbitrary

    Jean Bourgain. The sum-product theorem in Z /q Z with q arbitrary. Journal d'Analyse Math \'e matique , 106(1):1, 2008

    work page 2008

  8. [8]

    Varj \'u

    Jean Bourgain and P \'e ter P. Varj \'u . Expansion in SL _d( Z /q Z ),\,q arbitrary. Invent. Math. , 188(1):151--173, 2012

    work page 2012

  9. [9]

    Effective B \'ezout identities in Q [z_1,..., z_n]

    Carlos A Berenstein and Alain Yger. Effective B \'ezout identities in Q [z_1,..., z_n] . Acta Math. , 166(3):69--120, 1991

    work page 1991

  10. [10]

    Super-approximation, II : the p -adic case and the case of bounded powers of square-free integers

    Alireza Salehi Golsefidy. Super-approximation, II : the p -adic case and the case of bounded powers of square-free integers. Journal of the European Mathematical Society , 21(7):2163--2232, 2019

    work page 2019

  11. [11]

    S um-product phenomena: p -adic case

    Alireza Salehi Golsefidy. S um-product phenomena: p -adic case. Journal d'Analyse Math \'e matique , 142(2):349--419, 2020

    work page 2020

  12. [12]

    Quasirandom groups

    William T Gowers. Quasirandom groups. Combinatorics, Probability and Computing , 17(3):363--387, 2008

    work page 2008

  13. [13]

    Salehi Golsefidy and P \'e ter P

    A. Salehi Golsefidy and P \'e ter P. Varj \'u . Expansion in perfect groups. Geom. Funct. Anal. , 22(6):1832--1891, 2012

    work page 2012

  14. [14]

    Linear random walks on the torus

    Weikun He and Nicolas de Saxc\'e. Linear random walks on the torus. arXiv preprint arXiv: 1910.13421 , 2019

    work page arXiv 1910

  15. [15]

    Trou spectral dans les groupes simples

    Weikun He and Nicolas de Saxc \'e . Trou spectral dans les groupes simples. arXiv preprint arXiv:2103.06679 , 2021

    work page arXiv 2021

  16. [16]

    H. A. Helfgott. Growth and generation in SL _2( Z/p Z) . Ann. of Math. (2) , 167(2):601--623, 2008

    work page 2008

  17. [17]

    Explicit constructions of concentrators

    Grigorii Aleksandrovich Margulis. Explicit constructions of concentrators. Problemy Peredachi Informatsii , 9(4):71--80, 1973

    work page 1973

  18. [18]

    Sum-product phenomenon in quotients of rings of algebraic integer

    Jincheng Tang and Xin Zhang. Sum-product phenomenon in quotients of rings of algebraic integer. 2023

    work page 2023

  19. [19]

    Expansion in SL_d(O_K/I) , I square-free

    P \'e ter P Varj \'u . Expansion in SL_d(O_K/I) , I square-free. J. Eur. Math. Soc.(JEMS) , 14(1):273--305, 2012

    work page 2012