2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167132Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S) established in an earlier paper is sharp in this case. The geometric analysis underlying our result determines the conectivity properties of horospheres in thick Euclidean buildings.27 pagesGroup TheoryGeometric Topology20G30; 20E42; 20F65text