Introduction .................................................... ix
Acknowledgements ............................................... xii
Part 1. Positive Automorphisms ................................... 1
1.1 Van Kampen diagrams ......................................... 3
1.2 Singularities and bounded cancellation ...................... 7
1.3 Past, future and colour .................................... 10
1.4 Strategy, strata and conditioning .......................... 12
1.5 Preferred futures, fast letters and cancellation ........... 14
1.6 Counting non-constant letters .............................. 17
1.7 The bound on  |A4(S0, μ)| and |A2(S0, μ)| ......... 22
1.8 The pleasingly rapid consumption .of colours ............... 26
1.9. Teams and their associates ................................. 37
1.10 The Bonus Scheme ........................................... 51
1.11 The proof of Theorem С ..................................... 58
1.12 Glossary of constants ...................................... 59
Part 2 Train Tracks and the Beaded Decomposition ............... 61
2.1 Improved relative train track maps ......................... 63
2.2 Hard splittings ............................................ 68
2.3 A small reduction .......................................... 71
2.4 Nibbled futures ............................................ 72
2.5 Passing to an iterate of ƒ ................................. 8I
2.6 The nibbled futures of GEPs ................................ 83
2.7 Proof of the Beaded decomposition Theorem .................. 88
2.8 Refinements of the Beaded Decomposition Theorem ............ 88
Part 3 The General Case ........................................ 93
3.1 The structure of diagrams .................................. 95
3.2 Adapting diagrams to the beaded decomposition .............. 98
3.3 linear bounds on the length of corridors .................. 100
3.4 Replacing ƒ by a suitable iterate ......................... 100
3.5 Preferred futures of beads ................................ 103
3.6 Counting fast beads ....................................... 107
3.7 HNP-cancellation and reapers .............................. 110
3.8 Non-fast and unbounded beads .............................. 115
3.9 The pleasingly rapid disappearance of colours ............. 121
3.10 Teams ..................................................... 131
3.11 The Bonus Scheme .......................................... 138
3.12 Prom bead norm to length .................................. 139
3.13 Corridor length functions and bracketing .................. 141
3.14 On a result of Brinkmann .................................. 145
Bibliography ................................................... 149
Index .......................................................... 151
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