Zeta functions of complexes arising from PGL(3)

Abstract

In this paper we obtain a closed form expression of the zeta function Z(Xr,u) of a finite quotient Xr of the Bruhat-Tits building of PGL(3) over a nonarchimedean local field F by a discrete cocompact torsion-free subgroup Gamma of PGL3. Analogous to a graph zeta function, Z(Xr, u) is a rational function with two different expressions and it satisfies the Riemann hypothesis if and only if Xr is a Ramanujan complex. (C) 2014 Elsevier Inc. All rights reserved.