Expansion in ${\text{SL}}_2(\mathbb{Z}\mathrm{/}q\mathbb{Z})$ and Zaremba's conjecture

We establish an expansion theory for ${\text{SL}}_2(\mathbb{Z}\mathrm{/}q\mathbb{Z})$. Incorporating this into a framework recently developed by Shkredov, we confirm Zaremba’s conjecture.