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

Ilya Kapovich; Catherine Pfaff

doi:10.1142/S0218196715500186

paper

A train track directed random walk on $O u t (F_{r})$

arXiv:1409.8044 · doi:10.1142/S0218196715500186

Abstract

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

arXiv commentVarious small updates, incorporating the referee's comments; to appear in the International Journal of Algebra and Computation

A train track directed random walk on $O u t (F_{r})$

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A train track directed random walk on Out(Fr​)

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A train track directed random walk on $O u t (F_{r})$