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The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms
Martin R. Bridson, Mathematical Institute, Oxford, England, and Daniel Groves, University of Illinois at Chicago, IL

Memoirs of the American Mathematical Society
2009; 152 pp; softcover
Volume: 203
ISBN-10: 0-8218-4631-0
ISBN-13: 978-0-8218-4631-5
List Price: US$78
Individual Members: US$46.80
Institutional Members: US$62.40
Order Code: MEMO/203/955
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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

  • Positive automorphisms
  • Train tracks and the beaded decomposition
  • The General Case
  • Bibliography
  • Index
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