The quadratic isoperimetric inequality for mapping tori of free group automorphisms

We prove that if F is a finitely generated free group and φ is an automorphism of F then F oφ 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 oφZ 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 leastarea diagram is bounded by a constant times the perimeter of the diagram, where the constant depends only on φ. Our proof that such a constant exists involves a detailed analysis of the ways in which the length of a word w ∈ F can grow and shrink as one replaces w by a sequence of words wm, where wm is obtained from φ(wm−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. Received by the editor 10 October, 2006. 2000 Mathematics Subject Classification. 20F65, (20F06, 20E36, 57M07).