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.
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Bridson, M.R., Vogtmann, K. Abelian covers of graphs and maps between outer automorphism groups of free groups. Math. Ann. 353, 1069–1102 (2012). https://doi.org/10.1007/s00208-011-0710-z
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DOI: https://doi.org/10.1007/s00208-011-0710-z