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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 spectrumof 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 spectrumof those operators acting on $L^2(\mathcal{B})$. In this paper, we prove thatthe standard non-uniform arithmetic quotient $PGL_4(\mathbb{F}_q[t])\backslash\mathcal{B}_4$ of $PGL_4(F)$ is weakly Ramanujan.

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