Finiteness properties of soluble arithmetic groups over global function fields

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

2002-12-29

arxiv.org/abs/math/0212365v2

Geom. Topol. 8(2004) 611-644

Mathematics

Group Theory

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper15.abs.html

Scientific paper

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.

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