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

Abstract

Let [Formula: see text] 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.