Relative hyperbolicity of free-by-cyclic extensions

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