Limit trees for free group automorphisms: universality (Q6652260)

Let \(\phi: F \rightarrow F\) be an automorphism of a free group of finite rank \(F\). In the author's earlier work [Forum Math. Sigma 12, Paper No. e57, 30 p. (2024; Zbl 07846245)], \(\phi\) is associated with a real pretree with several interesting properties: \N\begin{itemize}\N\item[(1)] the tree has a minimal isometric action of the free group with trivial arc stabilizers; \N\item[(2)] there is a unique expanding dilation of the tree that represents \(\phi\); \N\item[(3)] the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by \(\phi\). \N\end{itemize}\NIn particular the action on the tree detects the automorphism's dominating exponential dynamics.\N\NIn the paper quoted above the author motivated the existence and uniqueness theorem of a limit pretree by describing it as a free group analogue to the Nielsen-Thurston theory for surface homeomorphisms (which in turn can be seen as the surface analogue to the Jordan canonical form for linear maps). In this paper he proves that his construction is canonical. He also characterizes all very small trees that admit an expanding homothety representing a given automorphism. In the final appendix, he proves a variation of Feighn-Handel's recognition theorem (see [\textit{M. Feighn}, \textit{M. Handel}, Groups Geom. Dyn. 5, No. 1, 39--106 (2011; Zbl 1239.20036)]) for atoroidal outer automorphisms.