Exotic limit sets of Teichmüller geodesics in the HHS boundary

Abstract

We answer a question of Durham, Hagen, and Sisto, proving that a Teichmüller geodesic ray does not necessarily converge to a unique point in the hierarchically hyperbolic space boundary of Teichmüller space. In fact, we prove that the limit set can be almost anything allowed by the topology. MSC 2010 Subject Classification: 30F60, 32Q05 (primary), 57M50 (secondary) IntroductionLet S = S g be a connected, closed, orientable surface of genus g ≥ 2, and let T (S) denote the Teichmüller space of S equipped with the Teichmüller metric. Masur [15] proved that T (S) is not non-positively curved in the sense of Busemann, and Masur and Wolf [20] showed that T (S) is not (Gromov) hyperbolic. In this paper, we explore to what extent T (S) has features of negative curvature by studying the asymptotic behavior of geodesics.The notion of Gromov boundary can be generalized in several ways to obtain a boundary for T (S). For example the visual boundary and the Morse boundary agree with the Gromov boundary when the space is hyperbolic, and these boundaries are well-defined for T (S) (see [21] and [5]). Here we consider another generalization. Both hyperbolic spaces and T (S) can be equipped with a geometric structure defined by Behrstock, Hagen, and Sisto [1] called a hierarchically hyperbolic space (HHS) structure. (That T (S) can be equipped with such a structure follows from the results in [7], [8], [19], [27].) These structures were used by Durham, Hagen, and Sisto [6] to construct a boundary, which we will call the HHS boundary (see Section 2 for definitions).Working in the HHS paradigm, the question becomes how do the asymptotics of geodesic rays in the HHS boundary of T (S) compare to those of geodesic rays in the HHS boundary of a hyperbolic space? The identity map on a hyperbolic space extends to a homeomorphism between its HHS and Gromov boundaries, so certainly in this case geodesic rays are wellbehaved. In [6] Durham, Hagen, and Sisto asked for a description of limit sets of Teichmüller geodesic rays in the HHS boundary. Our main result provides an answer to this question.Theorem 1.1. Given a continuous map γ : R → 2 to the standard 2-simplex, there exists a Teichmüller geodesic ray G in T (S 3 ) and an embedding of 2 into the HHS boundary of T (S 3 ) such that the limit set of G in the HHS boundary is the image of γ(R).The study of limiting behaviors of Teichmüller geodesic rays began with Kerckhoff [10]. He proved that the Teichmüller boundary of T (S) (the collection of all geodesic rays emanating from a fixed basepoint) is basepoint dependent. Since then, the limit sets of geodesic rays in Thurston's compactification of T (S) by PMF(S), the space of projectivized measured foliations, have received much attention. Masur [16] showed that almost all Teichmüller geodesic rays converge to a unique point in PMF. Lenzhen [12] provided the first example of a geodesic ray whose limit set in PMF is more than one 1 arXiv:1704.08645v1 [math.GT]

Boundaries and automorphisms of hierarchically hyperbolic spaces

“…Thus these special hierarchy ray representatives for boundary points are strictly necessary. See also work of Mousely [Mou19] for examples of more exotic Teichmüller rays and their limit sets in the HHS boundary of Teichmüller space, which takes advantage of the fact that Teichmüller geodesics are not hierarchy rays, but are very close to being so [Raf14].…”