  Let 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.
