Length functions, curvature, and the dimension of discrete groups

We work with the class of groups that act properly by semisimple isometries on complete CAT(0) spaces. Define dim(ss) Gamma to be the minimal dimensio n in which Gamma admits such an action. By examining the nature of translat ion length functions we show that there exist finitely-presented, torsion-f ree groups Gamma for which dim(ss) Gamma is greater than the cohomological dimension of Gamma. We also show that dim(ss) Gamma can decrease when one p asses to a subgroup of finite index.