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

Abstract

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