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The Group Fixed by a Family of Injective Endomorphisms of a Free Group
Warren Dicks, Universitat Autónoma de Barcelona, Spain, and Enric Ventura, Universitat Politécnica de Catalunya, Barcelona, Spain

Contemporary Mathematics
1996; 81 pp; softcover
Volume: 195
ISBN-10: 0-8218-0564-9
ISBN-13: 978-0-8218-0564-0
List Price: US$25
Member Price: US$20
Order Code: CONM/195
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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.


Graduate students and research mathematicians interested in finite group theory; also suitable as a supplementary text for combinatorial group theory courses.

Table of Contents

  • Groupoids
  • Measuring devices
  • Properties of the basic operations
  • Minimal representatives and fixed subgroupoids
  • Open problems
  • Bibliography
  • Index
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