Spectrum of the weighted adjacency operator on a nonuniform arithmetic quotient of PGL3

We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a P GLp3, F q rtsq quotient of r A 2 -type building.We prove that the set of non-trivial approximate eigenvalues pλ `, λ ´q of the weighted adjacency operators A w on the quotient induced from the colored adjacency operators A ˘on the building for P GL 3 contains the simultaneous spectrum of A ˘and another hypocycloid with three cusps.As a byproduct, we re-establish a proof of the fact that P GLp3, F q rtsqzP GLp3, F q ppt ´1qqq{P GLp3, F q rrt ´1ssq is not a Ramanujan complex, from a combinatorial aspect.