Limit pretrees for free group automorphisms: existence (Q6536613)

Let \(F\) be a free group and \(\phi \in \mathrm{Aut}(F)\). In this paper the author associates to \(\phi\) a trace a real pretree with several interesting properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretreeautomorphism of the real pretree that represents the free group automorphism. Finally, 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.\N\NThe main result is Theorem 3.3: If \(\phi: F \rightarrow F\) is an automorphism of a finitely generated free group, then there is: (1) a minimal rigid \(F\)-action on a real pretree \(T\) with trivial arc stabilizers; (2) a \(\phi\)-equivariant \(F\)-expanding pretree-automorphism \(f: T \rightarrow T\); and (3) an element in \(F\) is \(T\)-elliptic if and only if it grows polynomially under \([\phi]\)-iteration.\N\NThe non-trivial point stabilizers are finitely generated. Moreover, these subgroups are proper and have rank strictly less than that of \(F\) if and only if \([\phi]\) is exponentially growing.\N\NThis 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.