## Abstract

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element φ_{n} 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 → ∞. In this paper, we construct a natural' train track directed' random walk on Out(F_{r}) (where r ≥ 3). We show that, for the element φ_{n} ∈ Out(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" 3/2-r (so that the geometric index of the attracting tree T_{φ} of φ_{n} is 2r-3), has index list {3/2-r} and the ideal Whitehead graph being the complete graph on 2r-1 vertices, and that the axis bundle of φ_{n} in the Outer space CV_{r} consists of a single axis.

Original language | English (US) |
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Pages (from-to) | 745-798 |

Number of pages | 54 |

Journal | International Journal of Algebra and Computation |

Volume | 25 |

Issue number | 5 |

DOIs | |

State | Published - Aug 18 2015 |

## Keywords

- Fully irreducible
- free group automorphisms
- index theory
- random walks
- train track maps

## ASJC Scopus subject areas

- Mathematics(all)