Abstract
AbstractWe show that arithmetic lattices in $$\textrm{SL}_{2}(\mathbb {R})$$
SL
2
(
R
)
, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $$\mathbb {Q}$$
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.
Funder
Swiss Federal Institute of Technology Zurich
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference32 articles.
1. Casselman, W.: On some results of Atkin and Lehner. Math. Ann. 201, 301–314 (1973)
2. Chu, M., Li, H.: Small generators of cocompact arithmetic Fuchsian groups. Proc. Am. Math. Soc. 144(12), 5121–5127 (2016)
3. Chuman, Y.: Generators and relations of $$\Gamma _{0}(N)$$. J. Math. Kyoto Univ. 13, 381–390 (1973)
4. Eichler, M.: Lectures on Modular Correspondences, vol. 56. Tata Institute of Fundamental Research Bombay, Mumbai (1955)
5. Eichler, M.: The basis problem for modular forms and the traces of the Hecke operators. In: Modular Functions of One Variable, I (Proceedings of International Summer School, University of Antwerp, Antwerp, 1972). Lecture Notes in Mathematics, vol. 320, pp. 75–151. Springer, Berlin (1973)