On two-generator subgroups of mapping torus groups

We prove that if G_φ= F, t| t x t⁻¹ =φ(x), x∈ F is the mapping torus group of an injective endomorphism φ: F→ F of a free group F (of possibly infinite rank), then every two-generator subgroup H of G_φ is either free or a sub-mapping torus. As an application we show that if φ∈ Out(Fᵣ) (where r≥ 2) is a fully irreducible atoroidal automorphism then every two-generator subgroup of G_φ is either free or has finite index in G_φ.