Ramanujan bigraphs associated with SU(3) over a p-adic field

TL;DR

This paper investigates when quotients of a Bruhat--Tits tree related to SU(3) over a p-adic field are Ramanujan bigraphs, linking their properties to a Ramanujan conjecture and automorphic spectrum classification.

Contribution

It establishes a criterion for Ramanujan bigraphs associated with SU(3) over p-adic fields based on a Ramanujan type conjecture, extending known results from PGL(2).

Findings

A quotient of the Bruhat--Tits tree is Ramanujan if and only if G satisfies a Ramanujan type conjecture.

Classification of automorphic spectrum implies existence of infinite Ramanujan bigraph families.

Results are analogous to the PGL(2) case studied by Lubotzky-Phillips-Sarnak.

Abstract

We use the representation theory of the quasisplit form G of SU(3) over a p-adic field to investigate whether certain quotients of the Bruhat--Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with G (which is a biregular bigraph) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of PGL(2) considered by Lubotzky-Phillips-Sarnak. As a consequence, the classification by Rogawski of the automorphic spectrum of U(3) implies the existence of certain infinite families of Ramanujan bigraphs.