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 |
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 |
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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 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.
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Table of Contents
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Chapters
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Introduction
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1. Positive Automorphisms
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2. Train Tracks and the Beaded Decomposition
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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 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