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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
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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

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.