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The quadratic isoperimetric inequality for mapping tori of free group automorphisms

About this Title

Martin R. Bridson, Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, United Kingdom and Daniel Groves, University of Illinois at Chicago, 2851 S. Morgan St., Chicago, Illinois 60607-7045

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 203, Number 955
ISBNs: 978-0-8218-4631-5 (print); 978-1-4704-0569-4 (online)
Published electronically: August 27, 2009
MathSciNet review: 2590896
Keywords:Free-by-cyclic groups, automorphisms of free groups, isoperimetric inequalities, Dehn functions
MSC: Primary 20F65; Secondary 20E36, 20F06, 57M07

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Table of Contents


  • Introduction
  • Part 1. Positive automorphisms
  • Part 2. Train tracks and the beaded decomposition
  • Part 3. The general case


We 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.

Our 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 $t$. We 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 $\phi$. Our 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, we develop a refinement of the improved relative train track technology due to Bestvina, Feighn and Handel.

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