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

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 R-tree all of whose branch points are trivalent.

Random trees in the boundary of Outer space

“…The main proposition of this section is the following. It is deduced from facts about random walks on groups acting on hyperbolic space (mainly results of Maher-Tiozzo [MT14]) and the bounded geodesic image property for translates of the axis A, a result previously established by the authors [KMPT18].…”

“…Connections to previous work. In our previous work [KMPT18], we proved that with probability approaching 1 as n → ∞, the random outer automorphism w n is fully irreducible and its attracting/repelling trees T wn ± are trivalent and nongeometric. However, since such trees form a countable, and hence ν-measure zero, subset of ∂CV r , this provides no information about a ν-typical tree in ∂CV r .…”

“…The original definition of a principal outer automorphism ϕ ∈ Out(F r ) is given in terms of its ideal Whitehead graph [HM11] and the reader can find a complete definition in those terms in [AKKP18] or [KMPT18]. Rather than recall the original definition here, we collect the essential properties that we will need and give an alternative characterization.…”

“…Before turning to the proof of Proposition 5.2, we briefly discuss random walks and hyperbolic spaces. The reader can find additional details in [MT14] and a similar setup in [KMPT18]. We assume throughout that µ is a probability measure on G with finite support, although this condition is far stronger than what is needed in this section.…”

Taking the High-Edge Route Through Outer Space in Rank 3

“…In [AKKP19], it is shown that the axes of principal fully irreducible outer automorphisms share a certain "stability" property with principal pseudo-Anosov axes in Teichmüller space. This stability property is then used in [KMPT18] and [KMPT19] to not only prove which outer automorphisms are random walk generic, but to understand properties held by a typical (with respect to the harmonic measure) tree in the boundary of Outer space.…”