Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms
 
Martin R. Bridson Mathematical Institute, Oxford, England
Daniel Groves University of Illinois at Chicago, Chicago, IL
The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms
eBook ISBN:  978-1-4704-0569-4
Product Code:  MEMO/203/955.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms
Click above image for expanded view
The Quadratic Isoperimetric Inequality for Mapping Tori of Free Group Automorphisms
Martin R. Bridson Mathematical Institute, Oxford, England
Daniel Groves University of Illinois at Chicago, Chicago, IL
eBook ISBN:  978-1-4704-0569-4
Product Code:  MEMO/203/955.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2032009; 152 pp
    MSC: Primary 20; Secondary 57

    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.

  • Table of Contents
     
     
    • Chapters
    • Introduction
    • 1. Positive Automorphisms
    • 2. Train Tracks and the Beaded Decomposition
    • 3. The General Case
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2032009; 152 pp
MSC: Primary 20; Secondary 57

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.

  • Chapters
  • Introduction
  • 1. Positive Automorphisms
  • 2. Train Tracks and the Beaded Decomposition
  • 3. The General Case
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.