Teichmüller geodesics that do not have a limit in 𝒫ℳℱ

Lenzhen, Anna

doi:10.2140/gt.2008.12.177

Cited by 35 publications

(36 citation statements)

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“…In the finite-dimensional setting the existence of geodesic Teichmüller rays with limit sets larger than a single point was first demonstrated by Lenzhen in [13]. Such rays and their limit sets in P ML(S) were further investigated by Leininger-Lenzhen-Rafi [12] and Chaika-Masur-Wolf [4].…”

Section: The Main Resultsmentioning

confidence: 99%

Limits of Teichmüller geodesics in the Universal Teichmüller space

Hakobyan

Šarić

2018

Proc. London Math. Soc.

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A Thurston boundary of the universal Teichmüller space T(D) is the set of projective bounded measured laminations PMLbddfalse(double-struckDfalse) of double-struckD. We prove that each Teichmüller geodesic ray in T(D) converges to a unique limit point in the Thurston boundary of T(D) in the weak∗ topology. In particular, there is an open and dense set of geodesic rays, which have unique (weak*‐)limits in the Thurston boundary. We also show that the main result is sharp by providing an example of a Teichmüller geodesic, which does not converge in the uniform weak∗ topology.

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Section: The Main Resultsmentioning

confidence: 99%

Limits of Teichmüller geodesics in the Universal Teichmüller space

Hakobyan

Šarić

2018

Proc. London Math. Soc.

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“…In [12] Lenzhen gives an explicit formula for T i n and gives a useful bound for the extremal length of α i (n) along G.…”

Section: Teichmüller Geodesic Rays From Irrational Numbersmentioning

confidence: 99%

Exotic limit sets of Teichmüller geodesics in the HHS boundary

Mousley

2019

Groups Geom. Dyn.

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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]

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“…See also Theorem 3.17 for a more precise statement. Our construction closely follows that of Lenzhen [Len08] who gave the first examples of Teichmüller geodesics having 1-dimensional limit sets in the Thurston compactification.…”

Section: Introductionmentioning

confidence: 98%

“…A number of authors have studied the limiting behavior of Teichmüller geodesics in relation to the Thurston compactification of Teichmüller space, [Mas82,Ker80] [ Len08,LM10], [LLR13,CMW14], [BLMR16a,LMR16]. This work has highlighted the delicate relationship between the vertical foliation of the quadratic differential defining the geodesic and the limit set in the Thurston boundary.…”

Section: Introductionmentioning

confidence: 99%