Finiteness properties of solvable S-arithmetic groups over function fields
by
Kai-Uwe Bux
J.W. Goethe-University Frankfurt a.M. (Germany)

Let B be a Borel subgroup of a Chevalley group, e.g. the group of upper triangular matrices with determinant 1. Let K denote a global function field, e.g. the field k(t) of rational functions over a finite field k. For a finite non-empty set of primes S let O_S denote the corresponding S-arithmetic subring of K, e.g. the ring k[t] of polynomials (|S|=1) or the ring k[t, t^-1] of Laurent polynomials (|S|=2).

We will determine the finiteness properties of the groups B(O_S). It is known that in the number field case these groups are of type F_\infty. Using the action of B(O_S) on the product of the Bruhat-Tits buildings associated to the primes in S, we prove:

Theorem.  The group B(O_S) is of type F_|S|-1 but not of type FP_|S|.

Date received: March 16, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cabb-06.