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

TL;DR

This paper introduces the edge zeta function for weighted complexes related to PGL(3) and provides an explicit formula for counting closed cycles from geodesics in the associated building.

Contribution

It defines the edge zeta function for a specific non-uniform complex and derives an explicit rational formula for it, connecting group theory and geometric cycles.

Findings

Derived the edge zeta function as a rational function.

Obtained an exact formula for counting closed cycles from geodesics.

Applied truncation techniques to achieve the formula.

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 $\operatorname{PGL}(3,\mathbb{F}_q[t])\backslash\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))/\operatorname{PGL}(3,\mathbb{F}_q[\![t^{-1}]\!])$, arising from the group $\operatorname{PGL}_3$, 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.