Strongly contracting geodesics in Outer Space

Abstract

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

On the geometry of Outer space

“…(We remark that the term "simplicial completion" is used slightly differently in [1], though the paper proves that her notion agrees with the one used here.) There are several natural ways of topologizing CV n , but they are all equivalent to giving it the quotient topology which arises from its description as the disjoint union of open simplices modulo the above face relations.…”

“…More precisely, she defined a coarse projection from all of Outer space onto the axis and proved that the image of a ball sufficiently far away from the axis has uniformly bounded diameter [1]. Making sense of this takes a great deal of care and requires developing coarse versions of many techniques.…”

Relative Twisting in Outer Space

“…In spite of this anomaly, we will refer to d L as the Lipschitz metric on CV k . We remark that it is known that the minimal Lipschitz constant, taken over all continuous maps f : G 1 → G 2 such that f • ρ 1 is homotopic to ρ 2 , is achieved by a map that is linear on edges ( [1,23]). …”

“…In the terminology from the proof of Theorem 2.1 in [1], the loop t is the subgraph G f ⊂ G and it is a legal loop. The automorphism δ is an example of a Dehn twist automorphism (see Section 6); in this case corresponding to the Bass-Serre tree arising from the HNN-extension A, c 0 , t | t −1 ct = c 0 .…”

“…We will be concerned with two types of such paths: those which connect two points G 1 and G 2 in the interior CV k , and those which are axes of fully irreducible elements. The latter were studied by Algom-Kfir in [1].…”