### The quadratic isoperimetric inequality for mapping tori of free-group automorphisms II: The general case

#### Submitted

This is the third and last in a series of papers whose main goal is to prove that if $F$ is a finitely generated free group and $\phi$ is an automorphism, then the Dehn function of $F\rtimes_\phi\Bbb Z$ is either linear or quadratic.

In the first paper in this series we proved the theorem under the additional assumption that $\phi$ was positive. In the second paper we used the train-track technology of Bestvina-Feighn-Handel to prove a Beaded Decomposition Theorem that identifies an appropriate analogue in the setting of IRTT representatives for letters in the positive case. In the current paper, we follow the strategy of the proof from paper I and deal with a nexus of additional complications in the cancellation arguments to complete the proof in the general case.

The main result concerns the geometry of van Kampen diagrams, but we explain how the statement of this result can be distilled into a simple algebraic statement, the Bracketing Theorem , which lays bare the non-deterministic time bound on the word problem that is implicit in the main result.