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 dimension in which $\Gamma$ admits such an action. By examining the nature of translation length functions we show that there exist finitely-presented, torsion-free 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 passes to a subgroup of finite index.

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Published 1 January 2001