Limit trees for free group automorphisms: universality

Abstract

To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique expanding dilation of the tree that represents the free group automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism's dominating exponential dynamics. As a corollary, our previously constructed limit pretree that detects the exponential dynamics is canonical. We also characterize all very small trees that admit an expanding homothety representing a given automorphism. In the appendix, we prove a variation of Feighn--Handel's recognition theorem for atoroidal outer automorphisms.

arXiv comment58 pages, 3 figures. v2: 62 pages, extended main theorem from topmost to dominating laminations and characterized expanding trees; updated references, including Feighn--Handel's recognition theorem. v3: incorporated referee comments

References in corpus (6)

• Limit groups and groups acting freely on R^n-trees
• The Tits alternative for the automorphism group of a free product
• The dynamics and geometry of free group endomorphisms
• Stabilizers of $\mathbb{R}$-trees with free isometric actions of $F_N$
• Abelian subgroups of \Out(F_n)
• Limit pretrees for free group automorphisms: existence