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).