The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms

Download or read book The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms written by Martin R. Bridson and published by American Mathematical Soc.. This book was released on 2010-01-15 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.