A train track directed random walk on Out(F_r)

Abstract

Abstract. 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 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 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 n 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.

Ideal Whitehead Graphs in Out(F r )

“…[Pfa13] gives examples in each rank where this occurs. [KP15] then shows that this behavior is generic along a particular "train track directed" random walk. We use the tools developed in this paper to prove that, even along the "train track directed" random walk, one still obtains fully irreducibles with a cut vertex in their ideal Whitehead graph and hence more than one axis in their axis bundle.…”

“…The proof of Lemma 2.36 below (and also that of [KP15]), prompts a revision of the definition of a Nielsen path prevention sequence so that the procedure applied (introduced in [Pfa13, Proposition 5.2] ends in each case in the legal turn scenario of [Pfa13] Proposition 5.2(IIc) and so that taking powers is never required in the procedure. Finally, it only relies on a sequence being admissible in the sense of [KP15], and Definition 2.16 above. We call such a sequence a legalizing Nielsen path prevention sequence.…”

“…By [MP13], this gives an example in each rank of a fully irreducibles whose axis bundle is more than just a single axis. It further gives examples of "nongeneric" behavior in the sense of [KP15].…”

Lone axes in outer space

“…What we prove in Theorem 4.7 of Section 4 is a necessary and sufficient condition for an ageometric fully irreducible outer automorphism to have a unique axis. One may note that examples of fully irreducibles satisfying the conditions of Theorem 4.7 can be found in [Pfa13a] and [Pfa13b] and it was in fact proved later, in [KP14], that satisfying these conditions is generic along a particular "train track directed" random walk.…”

Long turns, INP’s and indices for free group automorphisms

“…We would also like to thank Ilya Kapovich for some valuable comments on indices of random automorphisms, in the context of his recent work [16] with Pfaff.…”