# The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms

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*Martin R. Bridson; Daniel Groves*

The authors prove that if \(F\) is a
finitely generated free group and \(\phi\) is an automorphism
of \(F\) then \(F\rtimes_\phi\mathbb Z\) satisfies a
quadratic isoperimetric inequality.

The authors' proof of this theorem rests on a direct study of the geometry
of van Kampen diagrams over the natural presentations of free-by-cylic
groups. The main focus of this study is on the dynamics of the time
flow of \(t\)-corridors, where \(t\) is the generator of
the \(\mathbb Z\) factor in \(F\rtimes_\phi\mathbb Z\)
and a \(t\)-corridor is a chain of 2-cells extending across a
van Kampen diagram with adjacent 2-cells abutting along an edge
labelled \(t\). The authors prove that the length of
\(t\)-corridors in any least-area diagram is bounded by a
constant times the perimeter of the diagram, where the constant
depends only on \(\phi\). The authors' proof that such a constant exists
involves a detailed analysis of the ways in which the length of a word
\(w\in F\) can grow and shrink as one replaces \(w\) by
a sequence of words \(w_m\), where \(w_m\) is obtained
from \(\phi(w_{m-1})\) by various cancellation processes. In
order to make this analysis feasible, the authors develop a refinement of the
improved relative train track technology due to Bestvina, Feighn and
Handel.

#### Table of Contents

# Table of Contents

## The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms

- Introduction ix10 free
- Part 1. Positive Automorphisms 114 free
- 1.1. Van Kampen diagrams 316
- 1.2. Singularities and bounded cancellation 720
- 1.3. Past, future and colour 1023
- 1.4. Strategy, strata and conditioning 1225
- 1.5. Preferred futures, fast letters and cancellation 1427
- 1.6. Counting non-constant letters 1730
- 1.7. The bound on S0|A4(S0,)| and S0|A2(S0,)| 2235
- 1.8. The pleasingly rapid consumption of colours 2639
- 1.9. Teams and their associates 3750
- 1.10. The Bonus Scheme 5164
- 1.11. The proof of Theorem C 5871
- 1.12. Glossary of constants 5972

- Part 2. Train Tracks and the Beaded Decomposition 6174
- Part 3. The General Case 93106
- 3.1. The structure of diagrams 95108
- 3.2. Adapting diagrams to the beaded decomposition 98111
- 3.3. Linear bounds on the length of corridors 100113
- 3.4. Replacing f by a suitable iterate 100113
- 3.5. Preferred futures of beads 103116
- 3.6. Counting fast beads 107120
- 3.7. HNP-cancellation and reapers 110123
- 3.8. Non-fast and unbounded beads 118131
- 3.9. The pleasingly rapid disappearance of colours 121134
- 3.10. Teams 131144
- 3.11. The Bonus Scheme 138151
- 3.12. From bead norm to length 139152
- 3.13. Corridor length functions and bracketing 141154
- 3.14. On a result of Brinkmann 145158

- Bibliography 149162
- Index 151164 free