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 x(phi) 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 Z factor in F x(phi) 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 is an element of 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.