We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of Out.F n / act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of Out.F n / are Morse, meaning that a quasi-geodesic with endpoints on the axis stays within a bounded distance from the axis. 20E05; 20E36, 20F65 IntroductionThere exists a striking analogy between the mapping class groups of surfaces, and the outer automorphism group Out.F n / of a rank n free group. At the core of this analogy lies Culler and Vogtmann's Outer Space X n [16], a contractible finite dimensional cell complex on which Out.F n / has a properly discontinuous action. Like Teichmüller space, Outer Space has an invariant spine on which the action is cocompact, making it a good topological model for the study of Out.F n /. Indeed, Outer Space has played a key role in proving theorems for Out.F n /, which were classically known for the mapping class group. For example, Bestvina and Feighn [6] show that Out.F n / is a virtual duality group by showing that the Borel-Serre bordification of Outer Space is 2n 5 connected at infinity. There have been three well studied metrics on Teichmüller space: the Teichmüller metric, the Weil-Petersson metric, and the Lipschitz metric. When the present work was conducted, the study of the geometry of Outer Space was still in its infancy (see Handel and Mosher [23] and Francaviglia and Martino [20]). Since then, Bestvina [4] has found a new proof of the classification theorem of outer automorphisms using the geometry of Outer Space.One would like to define a metric on Outer Space so that fully irreducible elements of Out.F n / (which are analogous to pseudo-Anosov elements in MCG.S/) act by hyperbolic isometries with meaningful translation lengths. But immediately one encounters a problem: it isn't clear whether to require the metric to be symmetric. To clarify, we follow the discussion in Handel and Mosher [24]. Consider the situation of a pseudo-Anosov map acting on Teichmüller space T .S g;p / with the Teichmüller metric d T . Associated to is an expansion factor and two measured foliations

We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm · L that induces d. There is an Out (F n )-invariant "potential" defined on Outer space such that when · L is corrected by d , the resulting norm is quasi-symmetric. As an application, we give new proofs of two theorems of Handel-Mosher, that d is quasi-symmetric when restricted to a thick part of Outer space, and that there is a uniform bound, depending only on the rank, on the ratio of logs of growth rates of any irreducible f ∈ Out (F n ) and its inverse.

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. Let φ ∈ Out(F n ) be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism φ determines a free-by-cyclic group Γ = F n φ Z, and a homomorphism α ∈ H 1 (Γ; Z). By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, α has an open cone neighborhood A in H 1 (Γ; R) whose integral points correspond to other fibrations of Γ whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen's Teichmüller polynomial that computes the dilatations of all outer automorphism in A.