A train track directed random walk on Out (Fr)

Ilya Kapovich, Catherine Pfaff

Research output: Contribution to journalArticlepeer-review


Several known results, by Rivin, Calegari-Maher and Sisto, show that an element φn Out(Fr), obtained after n steps of a simple random walk on Out(Fr), is fully irreducible with probability tending to 1 as n → ∞. In this paper, we construct a natural' train track directed' random walk on Out(Fr) (where r ≥ 3). We show that, for the element φn ∈ Out(Fr), obtained after n steps of this random walk, with asymptotically positive probability the element φn has the following properties: φn is an ageometric fully irreducible, which admits a train track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that φn has "rotationless index" 3/2-r (so that the geometric index of the attracting tree Tφ of φn 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 φn in the Outer space CVr consists of a single axis.

Original languageEnglish (US)
Pages (from-to)745-798
Number of pages54
JournalInternational Journal of Algebra and Computation
Issue number5
StatePublished - Aug 18 2015


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

ASJC Scopus subject areas

  • Mathematics(all)


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