We work with the class of groups that act
properly by semisimple isometries on complete
CAT(0) spaces.
Define $\dim_{ss} g$ to be the minimal dimension in which a group
$G$ admits such an action. By examining the nature of translation length
functions, we show
that there exist finitely-presented, torsion-free
groups $G$ for which $\dim_{ss} G$ is greater than
the cohomological dimension of $G$. We also show that
$\dim_{ss} G$ can decrease when one passes to a subgroup of finite index.