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

TL;DR

This paper proves that certain Cayley graphs derived from dense subgroups of  and b7  are expanders across various moduli, advancing understanding of super approximation.

Contribution

It establishes super approximation results for  and b7 , showing their Cayley graphs form expanders mod q.

Findings

Cayley graphs form a family of expanders for these groups.

The results hold for groups generated by finite symmetric sets.

The work extends super approximation to new group settings.

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.