We show how to construct, for each r ≥ 3, an ageometric, fully irreducible φ ∈ Out(F r ) whose ideal Whitehead graph is the complete graph on 2r − 1 vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible φ ∈ Out(F 3 ). The result is a first step to an Out(F r ) version of the Masur-Smillie theorem proving precisely which index lists arise from singular measured foliations for pseudo-Anosov mapping classes.In this paper we additionally give a method for finding periodic Nielsen paths and prove a criterion for identifying representatives of ageometric, fully irreducible φ ∈ Out(F r ) arXiv:1301.6645v2 [math.GR]

In this paper we propose an Outer space analogue for the principal stratum of the unit tangent bundle to the Teichmüller space T (S) of a closed hyperbolic surface S. More specifically, we focus on properties of the geodesics in Teichmüller space determined by the principal stratum. We show that the analogous Outer space "principal" periodic geodesics share certain stability properties with the principal stratum geodesics of Teichmüller space. We also show that the stratification of periodic geodesics in Outer space exhibits some new pathological phenomena not present in the Teichmüller space context.

Abstract. Handel and Mosher define the axis bundle for a fully irreducible outer automorphism in [HM11]. In this paper we give a necessary and sufficient condition for the axis bundle to consist of a unique periodic fold line. As a consequence, we give a setting, and means for identifying in this setting, when two elements of an outer automorphism group Out(F r ) have conjugate powers.

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]

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.

We prove that a "random" free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. In particular, we show that its attracting (and repelling) tree is a nongeometric R-tree all of whose branch points are trivalent.

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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 paper we determine an analog to the theorem for Out(F3). That is, we determine which index lists permitted by the [GJLL98] index sum inequality are achieved by fully irreducible outer automorphisms of the rank-3 free group.