Relative hyperbolicity of free-by-cyclic extensions

Given a finitely generated free group F of rank( F )≥ 3 , we show that the mapping torus of φ is (strongly) relatively hyperbolic if φ is exponentially growing. As a corollary of our work, we give a new proof of Brinkmann's theorem which proves that the mapping torus of an atoroidal outer automorphism is hyperbolic. We also give a new proof of the Bridson–Groves theorem that the mapping torus of a free group automorphism satisfies the quadratic isoperimetric inequality. Our work also solves a problem posed by Minasyan and Osin: the mapping torus of an outer automorphism is not virtually acylindrically hyperbolic if and only if φ has finite order.