The rhombic dodecahedron and semisimple actions of Aut(F_n) on CAT(0) spaces

Abstract

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT$(0)$ spaces. If $n\ge 4$ then each of the Nielsen generators of Aut$(F_n)$ has a fixed point. If $n=3$ then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated $\Z^4\subset Aut(F_3)$ leaves invariant an isometrically embedded copy of Euclidean 3-space on which it acts as a discrete group of translations with the rhombic dodecahedron as a fundamental domain. An abundance of actions of the second kind is described. Constraints on maps from Aut$(F_n)$ to mapping class groups and linear groups are obtained. If $n\ge 2$ then neither Aut$(F_n)$ nor Out$(F_n)$ is the fundamental group of a compact K\"ahler manifold.