Free-group automorphisms, train tracks and the beaded decomposition
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We study the automorphisms ϕof 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 ϕthat behaves very much like the realization on the rose of a positive automorphism. This resemblance is encapsulated in the 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 $F\rtimes_ϕ\mathbb Z$, with ϕpositive, to the general case. To illustrate the wider utility of our topological normal form, we provide a short proof that for every w in F, the function $n\mapsto |ϕ^n(w)|$ grows either polynomially or exponentially.
Version 2 is 35 pages. The section on growth contained an error pointed out by Gilbert Levitt, and has been removed. A corrected verion will appear in a subsequent paper. Otherwise, changes are mostly cosmetic
Version 2 is 35 pages. The section on growth contained an error pointed out by Gilbert Levitt, and has been removed. A corrected verion will appear in a subsequent paper. Otherwise, changes are mostly cosmetic