Finiteness properties of soluble arithmetic groups over global function fields

Abstract

Let $\mathcal{G}$ be a Chevalley group scheme and $\mathcal{ℬ}\le \mathcal{G}$ a Borel subgroup scheme, both defined over $\mathbb{Z}$. Let $K$ be a global function field, $S$ be a finite non-empty set of places over $K$, and ${\mathcal{O}}_S$ be the corresponding $S$–arithmetic ring. Then, the $S$–arithmetic group $\mathcal{ℬ}\left({\mathcal{O}}_S\right)$ is of type $F_{\left|S\right|-1}$ but not of type $F{P}_{\left|S\right|}$. Moreover one can derive lower and upper bounds for the geometric invariants ${\Sigma }^m\left(\mathcal{ℬ}\left({\mathcal{O}}_S\right)\right)$. These are sharp if $\mathcal{G}$ has rank $1$. For higher ranks, the estimates imply that normal subgroups of $\mathcal{ℬ}\left({\mathcal{O}}_S\right)$ with abelian quotients, generically, satisfy strong finiteness conditions.