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
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,