Small diameters and generators for arithmetic lattices in $$\textrm{SL}_2(\mathbb {R})$$ and certain Ramanujan graphs

We show that arithmetic lattices in $\mathrm{SL}_{2}(\mathbb{R})$, stemming from the proper units of an Eichler order indefinite quaternion algebra over $\mathbb{Q}$, admit a `small' covering set. In particular, we give bounds on diameter if quotient space is co-compact. Consequently, these small generators. Our techniques also apply to definite algebras where Ramanujan-strength certain Ramanujan graphs without use bound.