Relative hyperbolicity of free-by-cyclic extensions

TL;DR

This paper proves that free-by-cyclic groups with exponentially growing automorphisms are relatively hyperbolic, providing new examples of hyperbolic extensions and offering a new proof of a quadratic isoperimetric inequality.

Contribution

It establishes relative hyperbolicity for free-by-cyclic groups with exponential growth automorphisms and constructs new hyperbolic extension examples, answering open questions.

Findings

Free-by-cyclic groups are relatively hyperbolic when automorphisms are exponentially growing.

New examples of free-by-free hyperbolic extensions are constructed.

Provides a new proof of the Bridson-Groves quadratic isoperimetric inequality.

Abstract

Given a finite rank free group $\mathbb{F}$ of $\mathsf{rank}(\mathbb{F})\geq 3$, we show that the mapping torus of $\phi$ is (strongly) relatively hyperbolic if $\phi$ 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.