xii INTRODUCTION each φ(ai) ∈ F contains no inverses a−1. j On the rose (1-vertex graph) with directed edges labelled ai, one has a natural representative for any automorphism of F . The key feature of positive automorphisms is the fact that the positive iterates of this representative restrict to injections on each edge of the graph. Such maps are the prototypes for train-track representatives. This discussion suggests a strategy that one might follow in order to prove one Main Theorem: first, one should prove it in the case of positive automorphisms, relying on the simplifications afforded by the positivity hypothesis to confront the web of large-scale cancellation phenomena that must be understood if one is to have any chance of proving the theorem in general. Then, in the general case, one should attempt to follow the architecture of the proof in the positive case, using a suitably refined train-track description of the automorphism in place of the positivity assumption. We shall implement the two stages of this plan in Parts 1 and 3 of this monograph, respectively. Ultimately, this strategy works. However, in Part 3, in order to bring our plan to fruition we have to deal with myriad additional complexities arising from intricate cancellations that do not arise in the positive case. Roughly speaking, these additional complexities correspond to the fact that most free group automorphisms do not have train track representatives, only relative train track representatives. In Part 2 of this monograph, we refine the theory of improved relative train track maps due Bestvina, Feighn and Handel [4], so as to tease-out features that allow us to adapt the crucial arguments from Part 1. A vital ingredient in this approach is the identification of basic units that will play the role in the general case that single edges (letters) played in the positive case. To this end, we develop a theory of beads, whose claim to the role is clinched by the Beaded Decomposition Theorem 3.2.1. This theorem is the main objective of Part 2. Indeed we have gone to considerable lengths to distill the entire contribution of Part 2 to Part 3 into this single statement and the important technical refinement of it described in Addendum 2.0.1. We have done so in order that the reader who is willing to accept it as an article of faith may proceed directly from Part 1 to Part 3. The introduction to each part of the book contains a more detailed explanation of its contents. Acknowledgements The first author’s work was supported in part by Research Fellowships from the EPSRC of Great Britain and by a Royal Society Wolfson Research Merit Award. Much of this work was undertaken whilst he was a Professor at Imperial College London, from which he was granted two terms of sabbatical leave. The second author was supported in part by a Junior Research Fellowship at Merton College, Oxford, by a Taussky-Todd Instructorship and a Senior Research Fellowship at the California Institute of Technology and by NSF Grant DMS-0504251. We thank these organisations for their support. We also thank the anonymous referee for his careful reading and helpful comments.

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