Edge zeta function and closed cycles in the standard non-uniform complex from ${\mathrm{PGL}}_3$

Abstract

In this paper, we define the edge zeta function of weighted complex. We also present the formula for the edge zeta function of the standard non-uniform complex $\mathrm{PGL}(3,{\mathbb{F}}_q[t])\backslash \mathrm{PGL}(3,{\mathbb{F}}_q((t^{-1})))/\mathrm{PGL}(3,{\mathbb{F}}_q[[t^{-1}]])$, arising from the group ${\mathrm{PGL}}_3$, as a rational function. Applying trunction in a specific direction is one of the main ingredient. As a result, we obtain the exact formula for the number of closed cycles coming from geodesics in the building.