###
Abelian covers of
graphs and maps between outer automorphism groups
of free groups

#### Martin R. Bridson and Karen Vogtmann

#### Version of 14 July 2010. Lemma 2.4 removed 2 Dec 2010.

We explore the existence of homomorphisms
between outer automorphism groups of free groups $\out\to\outm$.
We prove that if $n>8$ is even and $n\neq m\le 2n$,
or $n$ is odd and $n\neq m\le 2n-2$, then all such homomorphisms
have finite image; in fact they factor
through ${\rm{det}}\colon \out\to \Z/2$.
In contrast, if $m=r^n(n-1)+1$ with $r$ coprime to
$(n-1)$,
then there exists an embedding $\out\hookrightarrow\outm$. In order
to prove this last statement, we
determine when the action of $\out$ by homotopy equivalences
on a graph of genus $n$ can be lifted to an action on a normal covering with
abelian Galois group.