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

TL;DR

This paper introduces a new chamber zeta function for complexes of groups, extending Ihara--Bass theory to higher-rank structures, and provides a determinant formula and explicit counting results for galleries in the Bruhat--Tits building of PGL(3).

Contribution

It defines the chamber zeta function for higher-rank complexes and establishes a determinant formula linking it to a chamber transfer operator, extending classical graph theory.

Findings

Chamber zeta function is rational in its complex parameter.

Determinant formula expresses the zeta function as a reciprocal of a characteristic polynomial.

Explicit counting formulas for closed gallery classes and spectral asymptotics.

Abstract

We introduce the \emph{chamber zeta function} for a complex of groups, defined via an Euler product over primitive tailless chamber galleries, extending the Ihara--Bass framework from weighted graphs to higher-rank settings. Let $\mathcal{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_{3}(F)$ for a non-archimedean local field $F$ with residue field $\mathbb{F}_{q}$. For the standard arithmetic quotient $\Gamma\backslash\mathcal{B}$ with $\Gamma=\mathrm{PGL}_{3}(\mathbb{F}_{q}[t])$, we prove an Ihara--Bass type \emph{determinant formula} expressing the chamber zeta function as the reciprocal of a characteristic polynomial of a naturally defined chamber transfer operator. In particular, the chamber zeta function is \emph{rational} in its complex parameter. As an application of the determinant formula, we obtain explicit counting results for closed gallery classes arising from tailless galleries in $\mathcal{B}$, including exact identities and spectral asymptotics governed by the chamber operator.