Helly-type theorems, CAT$(0)$ spaces, and actions of automorphism groups of free groups

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 $\mathrm{\Gamma }$ 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 $\mathrm{F}\mathrm{i}\mathrm{x}\mathrm{D}\mathrm{i}\mathrm{m}(\mathrm{\Gamma })+1$ 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 $\mathrm{F}\mathrm{i}\mathrm{x}\mathrm{D}\mathrm{i}\mathrm{m}(\mathrm{A}\mathrm{u}\mathrm{t}(F_n))\ge \lfloor 2n\mathrm{/}3\rfloor$.