eBook ISBN:  9781470405694 
Product Code:  MEMO/203/955.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
eBook ISBN:  9781470405694 
Product Code:  MEMO/203/955.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 203; 2009; 152 ppMSC: 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 freebycylic 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 2cells extending across a van Kampen diagram with adjacent 2cells abutting along an edge labelled \(t\). The authors prove that the length of \(t\)corridors in any leastarea 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 \(\phi(w_{m1})\) 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


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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 freebycylic 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 2cells extending across a van Kampen diagram with adjacent 2cells abutting along an edge labelled \(t\). The authors prove that the length of \(t\)corridors in any leastarea 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 \(\phi(w_{m1})\) 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