The Group Fixed by a Family

of Injective Endomorphisms of a Free Group

Warren Dicks and 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 ¢ be an injective endomor-

phism 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 ofF,

the rank of the

intersection

H

n

Fix(¢) is at most the rank of

H.

They deduce that the rank

of the free subgroup which consists of the elements

ofF

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.

ISBN 0-8218-0564-9

9 780821 805640