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 ₁, 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 ᵢ, then such a subgroup system can be used to construct a collection of peripheral subgroups relative to which, the extension of F F by the free group generated by sufficiently high powers of ϕ 1, ⋯ ϕ k ₁, 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).
Ghosh et al. (2026) studied this question.