The rhombic dodecahedron and semisimple actions of ${\rm{Aut}}(F_n)$ on CAT$(0)$ spaces

Martin R. Bridson

Version of 22 Feb 2011. To appear in the volume for the 60th birthday of Mike Davis

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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
$\Z^4\subset{\rm{Aut}}(F_n)$ leaves invariant an isometrically embedded copy
of Euclidean 3-space $\E^3\hookrightarrow X$ on which it acts discretely with the
rhombic dodecahedron as a fundamental 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 K\"ahler manifold.