Work

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

Year: 2023

Type: article

Abstract: We 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.

Source: The Ramanujan Journal

DOI: 10.1007/s11139-023-00725-1

Author: Raphael S. Steiner

Institution: ETH Zurich

Language: Latvian

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,790

Funders: National Science Foundation, Eidgenössische Technische Hochschule Zürich