#### Geometry & Topology, Vol. 8 (2004)
Paper no. 15, pages 611--644.

## Finiteness properties of soluble arithmetic groups over global function fields

### Kai-Uwe Bux

**Abstract**.
Let G be a Chevalley group scheme and B<=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 O_S be the corresponding
S-arithmetic ring. Then, the S-arithmetic group B(O_S) is of type
F_{|S|-1} but not of type FP_{|S|}. Moreover one can derive lower and
upper bounds for the geometric invariants \Sigma^m(B(O_S)). These are
sharp if G has rank 1. For higher ranks, the estimates imply that
normal subgroups of B(O_S) with abelian quotients, generically,
satisfy strong finiteness conditions.
**Keywords**.
Arithmetic groups, soluble groups, finiteness properties, actions on buildings

**AMS subject classification**.
Primary: 20G30.
Secondary: 20F65.

**DOI:** 10.2140/gt.2004.8.611

**E-print:** `arXiv:math.GR/0212365`

Submitted to GT on 10 April 2003.
(Revised 8 April 2004.)
Paper accepted 19 December 2004.
Paper published 12 April 2004.

Notes on file formats
Kai-Uwe Bux

Cornell University, Department of Mathemtics

Malott Hall 310, Ithaca, NY 14853-4201, USA

Email: bux_math_2004@kubux.net

URL: http://www.kubux.net

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