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(Fn) Out(Fm). 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: Out(Fn) Z/2. In contrast, if m = rn(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(Fn) Out(Fm). In order to prove this last statement, we determine when the action of Out(Fn) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.
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