Weak Ramanujan property of the standard non-uniform arithmetic quotient of $PGL_4$

TL;DR

This paper proves that the standard non-uniform arithmetic quotient of the affine building associated with PGL_4 over a field of formal series over a finite field exhibits the weak Ramanujan property, indicating optimal spectral behavior.

Contribution

It establishes the weak Ramanujan property for the specific non-uniform arithmetic quotient of PGL_4, extending understanding of spectral properties in higher rank groups.

Findings

The quotient is weakly Ramanujan.

Spectral bounds match those of the universal cover.

Advances spectral theory of non-uniform arithmetic quotients.

Abstract

Let $F$ be a field of formal series over a finite field and $\mathcal{B}_d$ be the affine building associated to $PGL_d(F)$. Given a lattice $\Gamma$ in $PGL_d(F)$, the complex arising as a quotient $\Gamma\backslash \mathcal{B}_d$ is called weakly Ramanujan if every non-tivial discrete simultaneous spectrum of the colored adjacency operators $A_1,A_2,\ldots,A_{d-1}$ acting on $L^2(\Gamma\backslash \mathcal{B}_d)$ is contained in the simultaneous spectrum of those operators acting on $L^2(\mathcal{B})$. In this paper, we prove that the standard non-uniform arithmetic quotient $PGL_4(\mathbb{F}_q[t])\backslash \mathcal{B}_4$ of $PGL_4(F)$ is weakly Ramanujan.