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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
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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$74
Individual Members: US$44.40
Institutional Members: US$59.20
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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