Length functions, curvature and the dimension of discrete groups

Martin R. Bridson

ETH Preprint, December 1999 --- to appear in Math. Res. Lett.

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.