Random outer automorphisms of free groups: Attracting trees and their singularity structures

TL;DR

This paper demonstrates that a typical free group outer automorphism exhibits a specific geometric structure with a union of triangles in its ideal Whitehead graph and a nongeometric, trivalent branch point attracting tree.

Contribution

It establishes that random free group outer automorphisms are ageometric fully irreducible with a union of triangles in their ideal Whitehead graph and nongeometric $ eal$-trees with trivalent branch points.

Findings

Most random outer automorphisms are ageometric fully irreducible.

Their attracting trees are nongeometric $ eal$-trees.

All branch points in these trees are trivalent.

Abstract

We prove that a "random" free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. In particular, we show that its attracting (and repelling) tree is a nongeometric $\mathbb R$-tree all of whose branch points are trivalent