###
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

- This is a .pdf file
(14 pages, no figures)
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.