The quadratic isoperimetric inequality for mapping tori of free group automorphisms

We prove that if $F$ is a finitely generated free group and $\phi$ is an automorphism of $F$ then $F\rtimes_\phi\mathbb{Z}$ satisfies a quadratic isoperimetric inequality.

Our proof of this theorem rests on a direct study of the geometry of van Kampen diagrams over the natural presentations of free-by-cylic groups. The main focus of this study is on the dynamics of the time flow of $t$-corridors, where $t$ is the generator of the $\mathbb{Z}$ factor in $F\rtimes_\phi\mathbb{Z}$ and a $t$-corridor is a chain of 2-cells extending across a van Kampen diagram with adjacent 2-cells abutting along an edge labelled $t$. We prove that the length of $t$-corridors in any least-area diagram is bounded by a constant times the perimeter of the diagram, where the constant depends only on $\phi$. Our proof that such a constant exists involves a detailed analysis of the ways in which the length of a word $w\in F$ can grow and shrink as one replaces $w$ by a sequence of words $w_m$, where $w_m$ is obtained from $\phi(w_{m-1})$ by various cancellation processes. In order to make this analysis feasible, we develop a refinement of the improved relative train track technology due to Bestvina, Feighn and Handel.