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Let $S\subset \text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$ be finitesymmetric and assume $S$ generates a group $G$ which is Zariski-dense in$\text{SL}_2\times \text{SL}_2$. We prove that the Cayley graphs $$\{\mathcalCay(G (\text{mod }q), S (\text{mod }q))\}_{q\in \mathbb Z_+}$$ form a family ofexpanders.

Super approximation for $\text{SL}_2(\mathbb Z/q\mathbb Z)\times \text{SL}_2(\mathbb Z/q\mathbb Z)$