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

Abstract

Let $S\subset {\text{SL}}_2(\mathbb{Z})\times {\text{SL}}_2(\mathbb{Z})$ or ${\text{SL}}_2(\mathbb{Z})⋉{\mathbb{Z}}^2$ 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 ${\text{SL}}_2(\mathbb{Z})⋉{\mathbb{Z}}^2$. We prove that the Cayley graphs $$\{\mathcal{C}ay(G(\text{mod }q),S(\text{mod }q)){\}}_{q\in \mathbb{Z}}$$ form a family of expanders.

Minor revision of previous version