Relative hyperbolicity of free-by-cyclic extensions

Abstract

Given a finite rank free group $\mathbb{F}$ of $\mathsf{rank}(\mathbb{F})\ge 3$, we show that the mapping torus of $\varphi$ is (strongly) relatively hyperbolic if $\varphi$ is exponentially growing. We combine our result with the work of Button-Kropholler to answer a question asked by Minasyan-Osin regarding the acylindrical hyperbolicity of such free-by-cyclic extensions. As an application we construct new examples of free-by-free hyperbolic extensions where the elements of the quotient group are not necessarily fully irreducible. We also give a new proof of the Bridson-Groves quadratic isoperimetric inequality theorem.

A new proof of the Bridson-Groves quadratic isoperimetric inequality theorem ( for mapping tori of free group automorphisms) has been added as an application. Some minor corrections and notation changes from earlier version