Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Abstract

Abstract. By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [MS93] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an Out(Fr) analog of the Masur-Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [HM11] give a strictly finer invariant in the analogous Out(Fr) setting, we determine which of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in Out(F3). IntroductionLet F r denote the free group of rank r and Out(F r ) its outer automorphism group. In this paper we prove realization results for an invariant dependent only on the conjugacy class (within Out(F r )) of the outer automorphism, namely the "ideal Whitehead graph." 1.1. Main result. A "fully irreducible" (iwip) outer automorphism is the most commonly used analogue to a pseudo-Anosov mapping class and is generic. An element φ ∈ Out(F r ) is fully irreducible if no positive power φ k fixes the conjugacy class of a proper free factor of F r .We give in Subsection 2.3 the exact Out(F r ) definition of an ideal Whitehead graph. For now, to give context, we remark that, for a pseudo-Anosov surface homeomorphism, the component of an ideal Whitehead graph coming from a foliation singularity is a polygon with edges corresponding to the lamination leaf lifts bounding a principal region in the universal cover [NH86].Handel and Mosher define in [HM11] a notion of an ideal Whitehead graph for a fully irreducible outer automorphism, a finite graph whose isomorphism type is an invariant of the conjugacy class of the outer automorphism. In this paper we investigate the extent to which the Out(F r ) situation is more complicated by giving a partial answer to a question posed by Handel and Mosher in [HM11]: Question 1.1. For each r ≥ 2, which isomorphism types of graphs occur as IW(φ) for a fully irreducible φ ∈ Out(F r )? Theorem. A. Exactly eighteen of the twenty-one connected, simplicial five-vertex graphs are the ideal Whitehead graph IW(φ) for a fully irreducible outer automorphism φ ∈ Out(F 3 ).The twenty-one connected, simplicial five-vertex graphs ([CP84]

A train track directed random walk on Out(Fr)

“…Note that if θ = [x → yx], then the fact that θ is an automorphism of F r implies that y = x ±1 . The following notion is based on the work of Pfaff [Pfa12,Pfa13a], although the terminology that we use here is slightly different. A sequence θ 1 , .…”

“…The only difference between the terminology used here and that used in [Pfa12], is that in the terminology of [Pfa12] a pINP prevention sequence is only required to prevent pINPs when an admissible sequence starting with a pINP prevention sequence composes to give a rotationless expanding irreducible train track map. Lemma 5.5 of [Pfa12] shows that p = θ 1 , . .…”

“…, θ 6 , then every power of t is an admissible sequence and there exist k ≥ 1 such that g t k is rotationless. Since t k starts with p, Lemma 5.5 of [Pfa12] applies to t k and implies that g t k = (g t ) k has no pINPs. However, by definition, a path in R r is a pINP for g t if and only if this path is a pINP for (g t ) k .…”

“…In the Out(F r ) setting, not only is the ideal Whitehead graph IW(ϕ) a finer invariant (c.f. [Pfa13a,Pfa13b]), but it provides further information about the behavior of lamination leaves at a singularity. It is again an invariant of the conjugacy class of ϕ, also invariant under taking positive powers of ϕ.…”

“…A key tool in establishing that ϕ n is fully irreducible is the train track criterion of full irreducibility obtained in [Pfa13a] (see Proposition 2.27 below); we also discuss a related criterion obtained in [Kap14] (see Proposition 2.26 below). We substantially rely on ideas and results of [Pfa12,Pfa13a,Pfa13b], although the exposition given in the present paper is almost completely self-contained.…”