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

TL;DR

This paper establishes bounds on the diameter and generators of arithmetic lattices in SL(2,R) derived from quaternion algebras, leading to insights into the structure of related Ramanujan graphs.

Contribution

It provides new bounds on diameters and generators for these lattices and applies techniques to Ramanujan graphs without relying on the classical Ramanujan bound.

Findings

Bounds on diameters of co-compact quotient spaces

Existence of small generators for the lattices

Ramanujan-strength bounds for certain Ramanujan graphs

Abstract

We show that arithmetic lattices in $\mathrm{SL}_{2}(\mathbb{R})$, 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.