Relative hyperbolicity of free extensions of free groups

We study relative hyperbolicity of free-by-free groups: Given a collection of outer automorphisms of the free group we give sufficient conditions for some power of these automorphisms to generate a free group so that the corresponding extension is relatively hyperbolic. Namely, if ϕ 1 , ⋯ ϕ k \phi _1, \cdots \phi _k is a collection of exponentially growing outer automorphisms with a common invariant subgroup system such that any conjugacy class in the complement of this system grows exponentially under iteration by all ϕ i \phi _i , then such a subgroup system can be used to construct a collection of peripheral subgroups relative to which, the extension of F \mathbb {F} by the free group generated by sufficiently high powers of ϕ 1 , ⋯ ϕ k \phi _1, \cdots \phi _k will be hyperbolic. Moreover, we show that our conditions are also necessary if we are given a free-by-free group with a relative hyperbolic structure where the generating outer automorphisms have cusp-preserving property (motivated by homeomorphisms of surfaces which preserve cusps).