The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms (Memoirs of the American Mathematical Society, January 2010) by Martin R. Bridson (PDF)

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The authors prove that if $F$ is a finitely generated free group and $phi$ is an automorphism of $F$ then $Frtimesphimathbb Z$ 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 $Frtimesphimathbb Z$ 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 $win F$ can grow and shrink as one replaces $w$ by a sequence of words $wm$, where $wm$ is obtained from $phi(wm-1)$ 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. (MEMO/203/955)

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