# Hyperbolicity of a semidirect product

Let F be a finitely generated free group and let $\gamma : F \rightarrow F$ be an automorphism. Is the semidirect product $F \rtimes \mathbb{Z}$ an hyperbolic group? where $\mathbb{Z}$ acts in F via $\gamma$.

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The Bestvina-Feighn combination theorem says that this is true if and only if $\gamma$ has no nontrivial periodic conjugacy classes. See
I have one more question, If $\gamma$ is induced by a diffeomorphism $f: M\rightarrow M$ where M is a compact 2 manifold with boundary and $\pi_1 (M)$ is isomorphic to F, then my initial demidirect product is hyperbolic? – Luis Jorge Nov 17 '11 at 19:47
The answer to this last question is that it never is (hyperbolic) -- in this case f fixes the boundary components (as a set), which easily gives you a torus, which is $\pi_1$-injective, and so you're not hyperbolic, by the easy half of Andy's answer. – Daniel Groves Dec 7 '11 at 22:48