Length functions, curvature, and the dimension of discrete groups - Bridson, M (2001)
We work with the class of groups that act properly by semisimple isometrics on complete CAT(0) spaces. Define dimss Γ to be the minimal dimension in which Γ admits such an action. By examining the nature of translation length functions we show that there exist finitely-presented, torsion-free groups Γ for which dimss Γ is greater than the cohomological dimension of Γ. We also show that dimss Γ can decrease when one passes to a subgroup of finite index.