Finiteness properties of soluble arithmetic groups over global function fields

Bux, Kai-Uwe¹

¹ Cornell University, Department of Mathemtics, Malott Hall 310, Ithaca, New York 14853-4201, USA

Geometry & topology, Tome 8 (2004) no. 2, pp. 611-644.

Résumé

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.

Détail

DOI : 10.2140/gt.2004.8.611

Keywords: arithmetic groups, soluble groups, finiteness properties, actions on buildings

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