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

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.