#### Volume 8, issue 2 (2004)

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Finiteness properties of soluble arithmetic groups over global function fields

### Kai-Uwe Bux

Geometry & Topology 8 (2004) 611–644
 arXiv: math.GR/0212365
##### Abstract

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

##### Keywords
arithmetic groups, soluble groups, finiteness properties, actions on buildings
Primary: 20G30
Secondary: 20F65