𝐴𝑢𝑡(𝐹)→𝐴𝑢𝑡(𝐹/𝐹”) is surjective for free group 𝐹 of rank ≥4

Bachmuth, Seymour; Mochizuki, Horace Y.

doi:10.1090/S0002-9947-1985-0805954-3

$\textrm {Aut}(F)\to \textrm {Aut}(F/F”)$ is surjective for free group $F$ of rank $\geq 4$
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by Seymour Bachmuth and Horace Y. Mochizuki PDF

Trans. Amer. Math. Soc. 292 (1985), 81-101 Request permission

Abstract:

In this article, it is shown that the group of automorphisms of the free metabelian group $\Phi (n)$ of rank $n \geqslant 4$ is not only finitely generated but in fact every automorphism of $\Phi (n)$ is induced by an automorphism of the free group of the same rank $n$. This contrasts sharply with the authors’ earlier result [4] that any set of generators of the group of automorphisms of the free metabelian group $\Phi (3)$ of rank $3$ contains infinitely many automorphisms which are not induced by an automorphism of the free group of rank $3$.

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