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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 $\text{SL}_2(\mathbbZ)\times \text{SL}_2(\mathbb Z)$ or $\text{SL}_2(\mathbb Z)\ltimes \mathbbZ^2$. 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 $\text{ASL}_2$