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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

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.