### Homomorphisms from Automorphism Groups of Free Groups

#### Martin R. Bridson and Karen Vogtmann

#### Final version September 2002. Now appeared in Bull. London
Math. Soc., 35 (2003), 785--792.

The automorphism group of a finitely generated
free group
is the normal closure
of a single element of order $2$.
If $m$ is less than $n$ then a homomorphism
$Aut(F_n)\to Aut(F_m)$
can have cardinality at most $2$. More generally,
this is true of homomorphisms from $\Aut(F_n)$ to any group
that does not contain an isomorphic copy of the symmetric
group $S_{n+1}$. Strong constraints are also obtained on maps to groups
that do not contain a copy of $W_n= (\Bbb Z/2)^n\rtimes S_n$,
or of $\Bbb Z^{n-1}$. These results place constraints on how $\Aut(F_n)$
can act. For example, if $n\ge 3$ then any action of $\Aut(F_n)$ on
the circle (by homeomorphisms) factors through $\text{\rm{det}}:Aut(F_n)
\to \Bbb Z_2$
no non-trivial actions on the
circle (by homeomorphisms).