  Let $S\subset \text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$ be finite
symmetric 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 $$\{\mathcal
Cay(G (\text{mod }q), S (\text{mod }q))\}_{q\in \mathbb Z_+}$$ form a family of
expanders.
