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Super approximation for $\text{SL}_2\times \text{SL}_2$ and $\text{ASL}_2$

Jincheng Tang, Xin Zhang

TL;DR

The paper proves that Zariski-dense subgroups of ${ m SL}_2(f Z) imes { m SL}_2(f Z)$ or ${ m SL}_2(f Z) times f Z^2$ have super-approximation, i.e., their Cayley graphs mod $q$ form expander families for all moduli. It develops a novel combination of sum-product techniques over finite quotients, a bounded-generation framework for simple factors, and a new gluing mechanism to lift spectral gaps from small to large moduli, including cases with denominators. The key contributions include a constructive propagation of density via lines and thick segments in ${ m sl}_2$, a two-factor gluing tool, and the verification of two representative open cases in the Salehi-Golsefidy–Varjú conjectures, with extensions to ${ m SL}_2(f Z)[1/q_0]$-type groups. The approach provides new analytic–combinatorial tools for thin groups in product and affine settings, with potential implications for explicit expander constructions and arithmetic dynamics.

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.

Super approximation for $\text{SL}_2\times \text{SL}_2$ and $\text{ASL}_2$

TL;DR

The paper proves that Zariski-dense subgroups of or have super-approximation, i.e., their Cayley graphs mod form expander families for all moduli. It develops a novel combination of sum-product techniques over finite quotients, a bounded-generation framework for simple factors, and a new gluing mechanism to lift spectral gaps from small to large moduli, including cases with denominators. The key contributions include a constructive propagation of density via lines and thick segments in , a two-factor gluing tool, and the verification of two representative open cases in the Salehi-Golsefidy–Varjú conjectures, with extensions to -type groups. The approach provides new analytic–combinatorial tools for thin groups in product and affine settings, with potential implications for explicit expander constructions and arithmetic dynamics.

Abstract

Let or be finite symmetric and assume generates a group which is a Zariski-dense subgroup or . We prove that the Cayley graphs form a family of expanders.
Paper Structure (18 sections, 33 theorems, 337 equations)

This paper contains 18 sections, 33 theorems, 337 equations.

Table of Contents

  1. Introduction
  2. Notations and methodology
  3. Sketch of proof for Proposition \ref{['2132']}
  4. Preliminaries on combinatorics
  5. Preliminaries on Random Walks
  6. Bounded generation for simple factors
  7. A large set of traces
  8. A large set of commutative matrices
  9. Applying sum-product
  10. Constructing a line
  11. Constructing a segment
  12. Proof of Proposition \ref{['1631']}
  13. A gluing technique for $\text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$
  14. The case $q^{J_1}\geq (q_3')^{\frac{1}{2}}$
  15. The case $q^{J_2}\geq (q_3')^{\frac{1}{2}}, q^{J_3}>(q^{J_2})^{\frac{1}{2}}$.
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $S$ be a finite symmetric set that generates a group $G$ which is a Zariski dense subgroup of ${\normalfont \text{SL}}_2(\mathbb Z)\times {\normalfont \text{SL}}_2(\mathbb Z)$ or ${\normalfont \text{SL}}_2(\mathbb Z)\ltimes \mathbb Z^2$, then $G$ has super approximation with respect to all posit

Theorems & Definitions (66)

  • Conjecture 1.1: Question 2, GV12
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.2
  • Remark 2.8
  • Remark 2.10
  • Remark 2.11
  • Proposition 3.1
  • Lemma 3.4
  • proof : Proof of Lemma \ref{['0955']}
  • ...and 56 more