There are many interesting and fruitful analogies between the group of isome-
tries of hyperbolic n-space
the mapping class group of a finite type surface
acting on Teichm¨ uller space, and the outer automorphism group Out(Fr) of a rank
r free group Fr acting on outer space Xr.
In all cases the space has a natural compactification, obtained by adding a
boundary at infinity to which the group action extends continuously. In all cases
there is a class of “hyperbolic” elements whose extended action has source–sink
dynamics, with a repeller and an attractor at infinity: loxodromic isometries of
pseudo-Anosov mapping classes of a surface [McC85]; and outer automorphisms
of Fr which are fully irreducible [LL03], meaning that no proper, nontrivial free
factor of Fr has a conjugacy class which is periodic under the action of the outer
and in Teichm¨ uller space, a “hyperbolic” element φ has a unique axis
in the usual metric sense, a properly embedded, oriented, φ-invariant geodesic line
which φ translates in the forward direction. The axis of φ depends only on the
ordered pair (repeller, attractor), and this dependence is natural with respect to
the group action: the axis is the unique geodesic line whose negative end converges
to the repeller and whose positive end converges to the attractor.
In Xr, however, it is not reasonable to expect such a nice notion of an axis for
a fully irreducible φ Out(Fr). There does not seem to be a natural candidate
for a metric on Xr, and certainly not one that has the unique geodesic properties
or Teichm¨ uller space. So among the many properly embedded lines in Xr
whose two ends converge to the repeller and the attractor of φ, it seems hard to
single out one of those lines in a natural way and call it the axis of φ. Nevertheless,
among all such lines we shall select in a natural manner a particular collection of
them, the union of which we call the “axis bundle” of φ, a subset Xr with
the following properties:
Properness: is a closed subset of Xr proper homotopy equivalent to R.
The negative end of limits in compactified outer space on the repeller,
and the positive end limits on the attractor.
Naturality: depends only on the ordered pair (repeller, attractor) in a
manner which is natural with respect to the action of Out(Fr).
We shall do this only in the case where φ is nongeometric, meaning that it does not
arise from a homeomorphism of a compact surface with boundary. The geometric
case, while conceptually much simpler and more well understood (see [BH92]), has
some peculiarities whose inclusion in our theory would overburden an already well
laden paper.
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