Homomorphisms from automorphism groups of free groups

Abstract

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 $W_n=(\mathbb{Z}/2{)}^n⋊S_n$, or of ${\mathbb{Z}}^{n-1}$. 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 $\text{det}:Aut(F_n)\to {\mathbb{Z}}_2$ .

10 Pages, to appear in J. London Math. Soc