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
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