2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66114We 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(R) has finiteness length |S|-1 in the function field case, and infinite finiteness length in the number field case.28 pages, 5 figuresGroup TheoryGeometric Topology20G30 (primary); 20F65 (secondary)text