Arithmetic subgroups of Chevalley group schemes over function fields I: quotients of the Bruhat-Tits building by $\{P\}$-arithmetic subgroups

TL;DR

This paper investigates the action of arithmetic subgroups of Chevalley group schemes over function fields on Bruhat-Tits buildings, revealing the orbit space structure as a CW-complex glued with sector chambers parametrized by geometric data.

Contribution

It provides a detailed description of the quotient of the Bruhat-Tits building by $ extbf{G}(A)$, including the structure and parametrization of the orbit space in terms of Picard groups and rank.

Findings

Orbit space is a CW-complex glued with sector chambers.

Sector chambers are parametrized by Picard group and rank.

Any rational sector face contains an embeddable subsector.

Abstract

Let $\mathbf{G}$ be a reductive Chevalley group scheme (defined over $\mathbb{Z}$). Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve over a field $\mathbb{F}$. Let $P$ be a closed point on $\mathcal{C}$. Let $A$ be the ring of functions that are regular outside $\lbrace P \rbrace$. The fraction field $k$ of $A$ has a discrete valuation $\nu=\nu_{P}: k^{\times} \rightarrow \mathbb{Z}$ associated to $P$. In this work, we study the action of the group $ \textbf{G}(A)$ of $A$-points of $\mathbf{G}$ on the Bruhat-Tits building $\mathcal{X}=\mathcal{X}(\textbf{G},k,\nu_{P})$ in order to describe the structure of the orbit space $ \textbf{G}(A)\backslash \mathcal{X}$. We obtain that this orbit space is the ``gluing'' of a closed connected CW-complex with some sector chambers. The latter are parametrized by a set depending on the Picard group of $\mathcal{C} \smallsetminus \{P\}$ and on the rank of $\mathbf{G}$. Moreover, we observe that any rational sector face whose tip is a special vertex contains a subsector face that embeds into this orbit space.