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.

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Kai-Uwe Bux
Cornell University, Department of Mathemtics
Malott Hall 310, Ithaca, NY 14853-4201, USA

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