Finiteness properties of soluble arithmetic groups over global function fields
arXiv:math/0212365 · doi:10.2140/gt.2004.8.611
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 Σ^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.
Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper15.abs.html