Stable Strata of Geodesics in Outer Space

Abstract

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.

Taking the High-Edge Route Through Outer Space in Rank 3

“…} is degenerate for some k. We call g transparent if each illegal turn is prenull, if each turn taken by g k (e) for some k ∈ N and e ∈ E(Γ) is taken by g(e), and if each g-periodic vertex and direction is in fact fixed by g. Notice that every self-map has a transparent power. This is a strengthening of the notion given in [AKKP19] and so in particular implies it.…”

“…Principal fully irreducible outer automorphisms (sometimes just called principal outer automorphisms) were introduced in [AKKP19] as analogues of pseudo-Anosov surface homeomorphisms with only 3-pronged singularities. As in [AKKP19], we call a fully irreducible outer automorphism ϕ ∈ Out(F r ) principal if IW (ϕ) is the disjoint union of 2r − 3 triangles. By the main theorem of [MP16], principal outer automorphisms are lone axis outer automorphisms.…”

“…Various methods for constructing fully irreducible outer automorphisms with an assortment of properties can be found, for example, in [CP10], [Pfa13], [Pfa15a], and [CL15]. We focus on so-called principal fully irreducible outer automorphisms, as introduced in [AKKP19] to emulate principal pseudo-Anosov surface homeomorphisms. Principal pseudo-Anosovs homeomorphisms were proved generic (in a random-walk sense) in [GM17] and play a fundamental role in the work of Masur, for example, in [Mas82].…”

“…Principal pseudo-Anosovs homeomorphisms were proved generic (in a random-walk sense) in [GM17] and play a fundamental role in the work of Masur, for example, in [Mas82]. The existence of at least one principal fully irreducible outer automorphism in each rank was proved in [AKKP19].…”

“…A fully irreducible outer automorphism (more specifically Stallings fold decompositions of train track representatives, see Definition 7) determine geodesics (called axes) in Outer space. In [AKKP19], it is shown that the axes of principal fully irreducible outer automorphisms share a certain "stability" property with principal pseudo-Anosov axes in Teichmüller space. This stability property is then used in [KMPT18] and [KMPT19] to not only prove which outer automorphisms are random walk generic, but to understand properties held by a typical (with respect to the harmonic measure) tree in the boundary of Outer space.…”