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The Group Fixed by a Family of Injective Endomorphisms of a Free Group
 
Warren Dicks Universitat Autóma de Barcelona, Barcelona, Spain
Enric Ventura Universitat Politécnica de Catalunya, Barcelona, Spain
The Group Fixed by a Family of Injective Endomorphisms of a Free Group
The Group Fixed by a Family of Injective Endomorphisms of a Free Group
eBook ISBN:  978-0-8218-7786-9
Product Code:  CONM/195.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
The Group Fixed by a Family of Injective Endomorphisms of a Free Group
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The Group Fixed by a Family of Injective Endomorphisms of a Free Group
The Group Fixed by a Family of Injective Endomorphisms of a Free Group
Warren Dicks Universitat Autóma de Barcelona, Barcelona, Spain
Enric Ventura Universitat Politécnica de Catalunya, Barcelona, Spain
eBook ISBN:  978-0-8218-7786-9
Product Code:  CONM/195.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
  • Book Details
     
     
    Contemporary Mathematics
    Volume: 1951996; 81 pp
    MSC: Primary 20;

    This monograph contains a proof of the Bestvina-Handel 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 self-contained, 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 Bestvina-Handel 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 Bestvina-Handel is translated into the language of groupoids, and many details previously left to the reader are meticulously verified in this text.

    Readership

    Graduate 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
  • 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: 1951996; 81 pp
MSC: Primary 20;

This monograph contains a proof of the Bestvina-Handel 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 self-contained, 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 Bestvina-Handel 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 Bestvina-Handel is translated into the language of groupoids, and many details previously left to the reader are meticulously verified in this text.

Readership

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
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.