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 $\gamma$.
The BestvinaFeighn combination theorem says that this is true if and only if $\gamma$ has no nontrivial periodic conjugacy classes. See MR1152226 (93d:53053) Bestvina, M.(1UCLA); Feighn, M.(1RTG2) A combination theorem for negatively curved groups. J. Differential Geom. 35 (1992), no. 1, 85–101. 

