The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms PDF
by Martin R Bridson
Description
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.
Information
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Download - Immediately Available
- Format:PDF
- Pages:152 pages
- Publisher:American Mathematical Society
- Publication Date:01/01/1900
- Category:
- ISBN:9781470405694
Information
-
Download - Immediately Available
- Format:PDF
- Pages:152 pages
- Publisher:American Mathematical Society
- Publication Date:01/01/1900
- Category:
- ISBN:9781470405694
