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

We show that arithmetic lattices in ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})$, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $\mathbb{Q}$, admit a `small’ covering set. In particular, we give bounds on the diameter if the quotient space is co-compact. Consequently, we show that these lattices admit small generators. Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.