eBook ISBN:  9780821877869 
Product Code:  CONM/195.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821877869 
Product Code:  CONM/195.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 195; 1996; 81 ppMSC: Primary 20;
This monograph contains a proof of the BestvinaHandel Theorem (for any automorphism of a free group of rank \(n\), the fixed group has rank at most \(n\) ) that to date has not been available in book form. The account is selfcontained, simplified, purely algebraic, and extends the results to an arbitrary family of injective endomorphisms.
Let \(F\) be a finitely generated free group, let \(\phi\) be an injective endomorphism of \(F\), and let \(S\) be a family of injective endomorphisms of \(F\). By using the BestvinaHandel argument with graph pullback techniques of J. R. Stallings, the authors show that, for any subgroup \(H\) of \(F\), the rank of the intersection \(H\cap \mathrm {Fix}(\phi )\) is at most the rank of \(H\). They deduce that the rank of the free subgroup which consists of the elements of \(F\) fixed by every element of \(S\) is at most the rank of \(F\).
The topological proof by BestvinaHandel is translated into the language of groupoids, and many details previously left to the reader are meticulously verified in this text.
ReadershipGraduate students and research mathematicians interested in finite group theory; also suitable as a supplementary text for combinatorial group theory courses.

Table of Contents

Chapters

Introduction

Chapter I: Groupoids

Chapter II: Measuring Devices

Chapter III: Properties of the Basic Operations

Chapter IV: Minimal Representatives and Fixed Subgroupoids

Open Problems

Bibliography

Index


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This monograph contains a proof of the BestvinaHandel Theorem (for any automorphism of a free group of rank \(n\), the fixed group has rank at most \(n\) ) that to date has not been available in book form. The account is selfcontained, simplified, purely algebraic, and extends the results to an arbitrary family of injective endomorphisms.
Let \(F\) be a finitely generated free group, let \(\phi\) be an injective endomorphism of \(F\), and let \(S\) be a family of injective endomorphisms of \(F\). By using the BestvinaHandel argument with graph pullback techniques of J. R. Stallings, the authors show that, for any subgroup \(H\) of \(F\), the rank of the intersection \(H\cap \mathrm {Fix}(\phi )\) is at most the rank of \(H\). They deduce that the rank of the free subgroup which consists of the elements of \(F\) fixed by every element of \(S\) is at most the rank of \(F\).
The topological proof by BestvinaHandel is translated into the language of groupoids, and many details previously left to the reader are meticulously verified in this text.
Graduate students and research mathematicians interested in finite group theory; also suitable as a supplementary text for combinatorial group theory courses.

Chapters

Introduction

Chapter I: Groupoids

Chapter II: Measuring Devices

Chapter III: Properties of the Basic Operations

Chapter IV: Minimal Representatives and Fixed Subgroupoids

Open Problems

Bibliography

Index