Finiteness Properties of S-Arithmetic Groups - a Survey

We give an overview about finiteness properties of soluble S-arithmetic groups. Both, the number field case and the function field case are covered. The main result is: If B is a Borel subgroup in a Chevalley group and R is an S-arithmetic ring, then the group B® has finiteness length |S|-1 in the function field case, and infinite finiteness length in the number field case.