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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2512.23276 |
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| _version_ | 1866914256743563264 |
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| author | Hong, Soonki Kwon, Sanghoon |
| author_facet | Hong, Soonki Kwon, Sanghoon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23276 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chamber zeta function and closed galleries in the standard non-uniform complex from $\operatorname{PGL}_3$ Hong, Soonki Kwon, Sanghoon Number Theory Combinatorics Dynamical Systems Primary 11F06, 20E42, Secondary 20C08, 05E18 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. |
| title | Chamber zeta function and closed galleries in the standard non-uniform complex from $\operatorname{PGL}_3$ |
| topic | Number Theory Combinatorics Dynamical Systems Primary 11F06, 20E42, Secondary 20C08, 05E18 |
| url | https://arxiv.org/abs/2512.23276 |