To any free group automorphism, we associate a real pretree with several nice
properties. First, it has a rigid/non-nesting action of the free group with
trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism
of the real pretree that represents the free group automorphism. Finally and
crucially, the loxodromic elements are exactly those whose (conjugacy class)
length grows exponentially under iteration of the automorphism; thus, the
action on the real pretree is able to detect the growth type of an element.
This construction extends the theory of metric trees that has been used to
study free group automorphisms. The new idea is that one can equivariantly blow
up an isometric action on a real tree with respect to other real trees and get
a rigid action on a treelike structure known as a real pretree. Topology plays
no role in this construction as all the work is done in the language of
pretrees.Comment: 41 pages, 1 figure. v2: 40 pages, referenced the seque