Fix(/3) is inert in
See Problem 2, and the commentary thereto, in the
Open Problems section of this article.
The paper is organized as follows.
In Chapter I we develop that part of the theory of groupoids we shall use. In
§I.l, we introduce a lot of tedious vocabulary that will allow us to express the Best-
vina-Handel topological argument in terms of groupoids, and so make transparent
the algebraic nature of their proof. In §1.2, we describe the operations that will be
performed on graphs, and the relationship with the free groupoids they generate,
and present Stallings' folding lemma which plays an important part throughout. In
§1.3, we review the basics concerning free groupoids, including sketches of proofs
of theorems by Reidemeister, Goldstein-Turner
and Nielsen-Schreier. In §1.4, we
introduce the notions of inertia for sets acted on by free groups, and for morphisms
of free groupoids, and for graph morphism immersions, all related to good behaviour
for intersections of subgroups of free groups. Here we sketch proofs of theorems by
Imrich-Neumann-Stallings and Magnus-Karrass-Solitar. In §1.5 we give a similar
analysis for the fixed subgroupoid of a groupoid endomorphism, sketching a proof
of a theorem by Goldstein-Turner-Cohen-Lustig-Bestvina-Handel.
Chapter II introduces the tools for measuring good behaviour of (continuous)
self-maps of graphs. In §ILl, we sketch Wielandt's elegant proof of the famous
Perron-Frobenius theorem, and describe the consequences which we shall use. In
§II.2, we introduce morphisms of finite (pseudo )metric graphs, prove the
ina-Feighn-Handel Bounded Cancellation Lemma, and describe the relation with
the Perron-Frobenius theorem. In §II.3, we describe five numerical quantities associ-
ated with a self-map of a finite graph, the most important being a Perron-Frobenius
Chapter III is quite a technical interlude checking that certain basic operations
can be used to lower, or at least not raise, various of the numerical quantities,
especially the Perron-Frobenius eigenvalue.
Chapter IV then studies self-maps of finite graphs which lexicographically min-
imize, within a similarity class, the five numerical quantities introduced in §2.3.
Such self-maps are called minimal representatives, and, in §4.1, it is shown that
every similarity class has a minimal representative. In §§IV.2-IV.4, various tremen-
dously useful properties of minimal representatives are obtained, and the implica-
tions for the fixed subgroupoid are developed; in essence, there exists an inductive
step in which the basis of the fixed subgroupoid increases by at most one edge. In
§IV.5, the main results are obtained by analysing this new edge. Thus, in Theo-
rem IV.5.4, we find that the fixed subgroupoid Fix(/3), of a rr1-injective groupoid
of a finitely generated, free groupoid G, has the property that
all groupoid morphisms equivalent to the inclusion Fix(/3)
G behave well with
respect to pullbacks.
We conclude with a selection of open problems.
Most of our presentation, especially §§IV.l-IV.4, consists of a faithful trans-
lation of arguments of Bestvina-Handel , and Gaboriau-Levitt-Lustig , into
the language of groupoids, which required a small amount of ingenuity. In spite of
our having found some simplifications, on balance the overall proof is now some-
what longer, because we provide many of the details that were originally left to the
reader. Let us emphasize then where our original contributions lie. Two of the five
numerical quantitities used are new, and this simplifies the original arguments, at
the price of obtaining less information which experts may miss, specifically about