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The super approximation property of $\mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z}) \times \mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z}) \times \mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z})$

Chong Zhang

Abstract

Take $S \subset \mathrm{SL}_2(\mathbb{Z}) \times \mathrm{SL}_2(\mathbb{Z})\times \mathrm{SL}_2(\mathbb{Z})$ be finite symmetric and assume $S$ generates a group $G$ which is Zariski-dense in $\mathrm{SL}_2 \times \mathrm{SL}_2\times \mathrm{SL}_2(\mathbb{Z})$. This paper proves that the Cayley graphs $$ \{\mathcal{C} a y(G(\bmod q), S(\bmod q))\}_{q \in \mathbb{Z}_{+}} $$ form a family of expanders.

The super approximation property of $\mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z}) \times \mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z}) \times \mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z})$

Abstract

Take be finite symmetric and assume generates a group which is Zariski-dense in . This paper proves that the Cayley graphs form a family of expanders.
Paper Structure (14 sections, 15 theorems, 143 equations)

This paper contains 14 sections, 15 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Sketch of the proposition 2.2
  4. Preliminary of combinatorics
  5. Random walks
  6. Bounded generation
  7. Gluing moduli
  8. The case $q_5^{\prime}>q_5^{\frac{1}{2}}$
  9. The case $q_5^{\prime \prime}>q_5^{\frac{1}{2}}, q_6>\left(q_6^{\prime \prime}\right)^{\frac{1}{2}}$
  10. The case $q_5^{\prime \prime}>q_5^{\frac{1}{2}}, q_6^{\prime}>\left(q_5^{\prime \prime}\right)^{\frac{1}{2}}$
  11. Proof of Proposition 2.2
  12. The case $q_l<q^{\frac{\epsilon}{2}}$
  13. The case $q^{\frac{\epsilon}{2}}<q_l<q^{1-\frac{\epsilon}{2}}$
  14. The case $q_l>q^{1-\frac{\epsilon}{2}}$

Key Result

Theorem 1.1

(Salehi-Golsefidy Gol19) Let $G<S L_n(\mathbb{Z})$ be finitely generated, then $G$ has the super approximation property with respect to bounded powers of square free integers if and only if the identity component $\mathbb{G}_0$ of the Zariski closure $\mathbb{G}$ of $G$ is perfect, i.e. $\left[\math

Theorems & Definitions (21)

  • Conjecture 1.0.1
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • ...and 11 more