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The quadratic isoperimetric inequality for

mapping tori of
free-group automorphisms II:

The general case

####
Martin R. Bridson and Daniel Groves

#### 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.