A train track directed random walk on $Out(F_r)$

TL;DR

This paper introduces a new 'train-track directed' random walk on Out(F_r) and shows that, with positive probability, elements obtained are fully irreducible with specific geometric and combinatorial properties, extending understanding of random elements in Out(F_r).

Contribution

The paper constructs a natural new random walk on Out(F_r) and demonstrates that it produces elements with detailed geometric and combinatorial properties with positive probability.

Findings

Elements are fully irreducible with positive probability.

Elements have a train-track representative with no periodic Nielsen paths.

The axis bundle of these elements in Outer space is a single axis.

Abstract

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element $\phi_n\in Out(F_r)$, obtained after $n$ steps of a simple random walk on $Out(F_r)$, is fully irreducible with probability tending to 1 as $n\to\infty$. In this paper we construct a natural "train-track directed" random walk $\mathcal W$ on $Out(F_r)$ (where $r\ge 3$). We show that, for the element $\phi_n\in Out(F_r)$, obtained after $n$ steps of this random walk, with asymptotically positive probability the element $\phi_n$ has the following properties: $\phi_n$ is an ageometric fully irreducible, which admits a train-track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that $\phi_n$ has "rotationless index" $\frac{3}{2}-r$ (so that the geometric index of the attracting tree $T_{\phi_n}$ of $\phi_n$ is $2r-3$), has index list $\{\frac{3}{2}-r\}$ and the ideal Whitehead graph being the complete graph on $2r-1$ vertices, and that the axis bundle of $\phi_n$ in the Outer space $CV_r$ consists of a single axis.