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

Soonki Hong, Sanghoon Kwon

TL;DR

This work defines and analyzes the type-1 chamber zeta function for the standard non-uniform arithmetic quotient $ extGammaackslasholdsymbol{eta}$ of the Bruhat–Tits building of $ ext{PGL}_3(F)$. It extends the Ihara–Bass framework to a higher-rank, non-uniform setting by introducing a chamber transfer operator $T$ on type-1 pointed chambers and proving a determinant formula $Z_ extGamma(u)=igl( ext{det}(I-uT)igr)^{-1}$, establishing rationality of the zeta function. The authors provide explicit counting results for closed gallery classes, derive the generating function for the weighted counts $N_m$, and give a finite-rank truncation analysis that yields a concrete rational expression for $Z_ extGamma(u)$. They also discuss spectral and Hecke-theoretic perspectives, highlighting potential connections to automorphic $L$-functions and suggesting avenues for generalization to higher rank and other non-uniform complexes.

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 $Γ\backslash\mathcal{B}$ with $Γ=\mathrm{PGL}_{3}(\mathbb{F}_{q}[t])$, 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.

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

TL;DR

This work defines and analyzes the type-1 chamber zeta function for the standard non-uniform arithmetic quotient of the Bruhat–Tits building of . It extends the Ihara–Bass framework to a higher-rank, non-uniform setting by introducing a chamber transfer operator on type-1 pointed chambers and proving a determinant formula , establishing rationality of the zeta function. The authors provide explicit counting results for closed gallery classes, derive the generating function for the weighted counts , and give a finite-rank truncation analysis that yields a concrete rational expression for . They also discuss spectral and Hecke-theoretic perspectives, highlighting potential connections to automorphic -functions and suggesting avenues for generalization to higher rank and other non-uniform complexes.

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 be the Bruhat--Tits building of for a non-archimedean local field with residue field . For the standard arithmetic quotient with , 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 , including exact identities and spectral asymptotics governed by the chamber operator.
Paper Structure (11 sections, 10 theorems, 87 equations, 6 figures)

This paper contains 11 sections, 10 theorems, 87 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Building associated to $\operatorname{PGL}(3,F)$
  3. Building $\mathcal{B}$ for $\operatorname{PGL}_3$
  4. Tailless galleries
  5. Galleries in non-uniform quotient
  6. Fundamental domain
  7. Admissible galleries in the quotient $\Gamma \backslash \mathcal{B}$
  8. Chamber zeta function and determinant formula
  9. Proof of Theorem \ref{['thm:1.1']}
  10. The determinant of $I-uT_k$
  11. The determinant of $I-uT$

Key Result

Theorem 1.1

For $G=\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))$ and $\Gamma=\operatorname{PGL}(3,\mathbb{F}_q[t])$, the type 1 chamber zeta function $Z_{\Gamma}(u)$ converges for sufficiently $u$, and it is given by

Figures (6)

  • Figure 1: A schematic position of this work among edge and chamber zeta functions for cocompact and non-uniform quotients.
  • Figure 2: Type of chambers
  • Figure 3: Type $1$ tailless pointed chamber gallery
  • Figure 4: The fundamental domain for $\Gamma\backslash\mathcal{B}$
  • Figure 5: $c_{m,n,i}$
  • ...and 1 more figures

Theorems & Definitions (19)

  • Theorem 1.1
  • proof : Idea of Proof
  • Remark 1.2: Idea of the traceable argument
  • Corollary 1.3: Counting closed galleries
  • Remark 1.4: Position within the higher-rank zeta program
  • Lemma 2.1
  • Lemma 3.1: Finiteness of admissible closed gallery classes
  • proof
  • Lemma 4.1
  • proof
  • ...and 9 more