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Free-group automorphisms, train tracks,

and the Beaded Decomposition

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Martin R. Bridson and Daniel Groves

#### Submitted

We study the automorphisms $\phi$ of a finitely generated
free group $F$. Building on the train-track
technology of Bestvina, Feighn and Handel, we provide a topological
representative $f:G\to G$ of a power of $\phi$ that behaves very much
like the realization on the rose of a positive automorphism. This
resemblance is encapsulated in the {\em Beaded Decomposition Theorem}
which describes the structure of paths in $G$ obtained by repeatedly
passing to $f$-images of an edge and taking subpaths.
This decomposition is the key
to adapting our proof of the quadratic isoperimetric inequality for
$F\rtimes_\phi\mathbb Z$, with $\phi$ positive, to the general case.