We prove a variety of fixed-point theorems for groups acting on CAT$(0)$ spaces. Fixed points are obtained by a bootstrapping technique, whereby increasingly large subgroups are proved to have fixed points: specific configurations in the subgroup lattice of $\Gamma$ are exhibited and Helly-type theorems are developed to prove that the fixed-point sets of the subgroups in the configuration intersect. In this way, we obtain lower bounds on the smallest dimension ${ \rm{ FixDim}}(\Gamma)+1$ in which various groups of geometric interest can act on a complete CAT$(0)$ space without a global fixed point. For automorphism groups of free groups, we prove ${ \rm{ FixDim}}({ \rm{ Aut}}(F_n)) \ge \lfloor 2n/3\rfloor$.
Categories: math.GR math.GT math.MG