Weak Ramanujan property of the standard non-uniform arithmetic quotient of $PG{L}_4$

Abstract

Let $F$ be a field of formal series over a finite field and ${\mathcal{B}}_d$ be the affine building associated to $PG{L}_d(F)$. Given a lattice $\Gamma$ in $PG{L}_d(F)$, 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 $A_1,A_2,\dots ,A_{d-1}$ 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 $PG{L}_4({\mathbb{F}}_q[t])\backslash {\mathcal{B}}_4$ of $PG{L}_4(F)$ is weakly Ramanujan.

arXiv comment37pages