# The Group Fixed by a Family of Injective Endomorphisms of a Free Group

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*Warren Dicks; Enric Ventura*

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.