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.