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

Abstract

We show that arithmetic lattices in textrm{SL}_{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.