2014/09/30 by Ilya Kapovich, Catherine Pfaff · 1 citation
Mathematics · #math.GR #math.GT #msc:20F65 #msc:57M
paper · pdf · doi:10.1142/s0218196715500186
Various small updates, incorporating the referee's comments; to appear in the International Journal of Algebra and Computation
arxiv created 2015/05/05 · arxiv updated 2015/06/02
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 \mathcal W 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 Tϕn 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.