Weak Ramanujan property of the standard non-uniform arithmetic quotient of PGL4

Let F be a field of formal series over a finite field and Formula: see text be the affine building associated to Formula: see text. Given a lattice Formula: see text in Formula: see text, the complex arising as a quotient Formula: see text is called weakly Ramanujan if every nontrivial discrete simultaneous spectrum of the colored adjacency operators Formula: see text acting on Formula: see text is contained in the simultaneous spectrum of those operators acting on Formula: see text. In this paper, we prove that the standard non-uniform arithmetic quotient Formula: see text of Formula: see text is weakly Ramanujan.