  We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in
a noncommutative isotropic absolutely almost simple group $G$ over a global
function field is one less than the sum of the local ranks of $G$ taken over
the places in $S$. This determines the finiteness properties for arithmetic
subgroups in isotropic reductive groups, confirming the conjectured finiteness
properties for this class of groups. Our main tool is Behr-Harder reduction
theory which we recast in terms of the metric structure of euclidean buildings.

Group Theory;Geometric Topology