Naturality means that for any nongeometric, fully irreducible φ, φ Out(Fr)
and any ψ Out(Fr), if ψ takes the repeller–attractor pair of φ to the repeller–
attractor pair of φ , then ψ takes to . One consequence of naturality is
that is invariant under φ, because the repeller–attractor pair of φ is invariant
under φ. Another is that axes are indifferent to iterates (positive powers) of φ: φn
and φ have the same axis for all n 0, because φn has the same repeller and the
same attractor as φ.
Certain differences between axes in hyperbolic space or Teichm¨ uller space on
the one hand, and axis bundles in outer space on the other hand, may already
be apparent from the above features. First, the topological relation between the
axis bundle and the real line is not homeomorphism but instead proper homotopy
equivalence. Moreover, there does not exist an upper bound depending only on r
for the “girth” or cross-sectional diameter of the axis bundle of a fully irreducible
φ Out(Fr), as we show by explicit construction in Section 7.1. Second, axes in
or Teichm¨ uller space are indifferent to inverses the axis of
is the orientation
reversal of the axis of φ. But as we have argued in [HM06], it seems unnatural to
insist that axes in Xr be indifferent to inverses, due to the phenomenon that a fully
irreducible φ Out(Fr) can have an expansion factor which is different than the
expansion factor of its inverse. Although our paper [HM07] offers a bare hint of an
“indifference to inverses” property, by bounding the ratio of the logarithms of the
expansion factors of φ and
nevertheless we do not attempt here to construct
a notion of axis which is indifferent to inverses.
The main results of this work are a theorem which proves equivalence of several
characterizations of the axis bundle from some of which the naturality feature of
the axis bundle is evident and a theorem which proves the properness feature of
the axis bundle. In the remainder of this introduction we shall describe the three
characterizations of the axis bundle, and the methods of proof.
1.1. Characterizations of the axis bundle. We shall characterize the axis
bundle by presenting three different definitions, the equivalence of which is one of
our major theorems. These definitions are motivated by various aspects of the the-
ory of outer automorphisms, and by analogies with various properties of Teichm¨uller
Fold lines. Fold lines form a natural class of paths in outer space arising from
several concepts: train track maps due to Bestvina and Handel [BH92]; Stallings
folds [Sta83]; and semiflow concepts behind Skora’s method for investigating ho-
motopy properties of spaces of R-trees ([Sko], and see also [Whi93], [Cla05],
Recall that elements of compactified outer space X
= Xr ∪∂Xr are represented
by minimal, very small actions of Fr on R-trees, what we shall call “Fr-trees”.
Two Fr-trees represent the same element of X
if and only if they are related
by a homothety (a bijection that multiplies the metric by a constant) that is Fr-
equivariant. An element of Xr itself is represented by a simplicial Fr-tree on which
the action of Fr is free. Passing to the quotient modulo Fr of a free, simplicial
Fr-tree, an element of outer space is also represented by a “marked graph”, a
metric graph G without valence 1 vertices whose fundamental group is marked by
an isomorphism Fr π1(G); while we allow G to have valence 2 vertices, their
presence or absence does not effect the element of Xr that G represents. The group
Out(Fr) acts on X
from the right: the image under φ Out(Fr) of a marked
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