The rhombic dodecahedron and semisimple actions of ${\rm{Aut}}(F_n)$ on {\rm CAT}$(0)$ spaces
Volume 214 / 2011
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 ${\rm Aut}(F_n)$ 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 $\mathbb Z^4\subset\mathop{\rm Aut} (F_3)$ leaves invariant an isometrically embedded copy of Euclidean 3-space $\mathbb E^3\hookrightarrow X$ on which it acts as a discrete group of translations with the rhombic dodecahedron as a Dirichlet domain. An abundance of actions of the second kind is described.
Constraints on maps from ${\rm Aut}(F_n)$ to mapping class groups and linear groups are obtained. If $n\ge 2$ then neither ${\rm Aut}(F_n)$ nor ${\rm{Out}}(F_n)$ is the fundamental group of a compact Kähler manifold.
