Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space II

Miyachi, Hideki

doi:10.1007/s10711-012-9727-z

Cited by 21 publications

(30 citation statements)

References 20 publications

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“…The above theorems have the following geometric interpretation. The uniqueness part if this theorem was proved independently by Miyachi [20]. We have seen that the set of Busemann points may be identified with the set of unit-area quadratic differentials at the basepoint.…”

Section: Introductionmentioning

confidence: 91%

The asymptotic geometry of the Teichmüller metric

Walsh

2018

Geom Dedicata

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We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the set of Busemann points. We show that the Busemann points are exactly the limits of the Teichmüller rays, and we give a necessary and sufficient condition for a sequence of Busemann points to converge to a Busemann point. Finally, we determine the detour metric on the boundary.

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Section: Introductionmentioning

confidence: 91%

The asymptotic geometry of the Teichmüller metric

Walsh

2018

Geom Dedicata

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“…Suppose on the contrary that i(a, F ) = 0. Then, by Proposition 4 in [31], {Q n } n∈N is a stable sequence in the sense that the set of accumulation points of {e −tn L F,yn } n∈N in the space of geodesic currents is contained in MF − {0} (as geodesic currents). In addition, any accumulation point L ∞ ∈ MF − {0} satisfies…”

Section: Theorem 72 (Uniqueness Of the Underlying Foliations) For Anymentioning

confidence: 99%

“…We refer to the argument in §5.3 of [31] (see also [16] and [26]). Let Γ G be the critical vertical graph of the holomorphic quadratic differential of J G,x .…”

Section: Structure Of the Null Setsmentioning

confidence: 99%

Geometry of the Gromov product: Geometry at infinity of Teichmüller space

Miyachi¹

2017

J. Math. Soc. Japan

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Abstract. This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a "coarsification" of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller space does not admit (quasi)-invertible rough-homothety.

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“…In the Gardiner-Masur boundary, simple closed curves and uniquely ergodic measured foliations are rigid in the following sense. Lemma 9.1 (Theorem 3 of [29]). Let p ∈ cl GM (T g,m ).…”

Section: Isometric Action On Teichmüller Spacementioning

confidence: 99%

“…[18] and Proposition 4.9 in [33]). Notice from Theorem 1.1 in [29] that the limits of two different Teichmüller rays emanating from x 1 are different in the Gardiner-Masur compactification. Hence, the horizontal and vertical foliations of corresponding quadratic differential q i should be λ i and λ i+2 for i = 1, 2.…”

Section: Proof Of Corollarymentioning

confidence: 99%

Unification of extremal length geometry on Teichmüller space via intersection number

Miyachi

2014

Math. Z.

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In this paper, we give a framework for the study of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur. There is a natural compactification using extremal length geometry introduced by Gardiner and Masur. The compactification is realized in a certain projective space. We develop the extremal length geometry in the cone which is defined as the inverse image of the compactification via the quotient mapping. The compactification is identified with a subset of the cone by taking an appropriate lift. The cone contains canonically the space of measured foliations in the boundary. We first extend the geometric intersection number on the space of measured foliations to the cone, and observe that the restriction of the intersection number to Teichmüller space is represented by an explicit formula in terms of the Gromov product with respect to the Teichmüller distance. From this observation, we deduce that the Gromov product extends continuously to the compactification. As an application, we obtain an alternative approach to a characterization of the isometry group of Teichmüller space. We also obtain a new realization of Teichmüller space, a hyperboloid model of Teichmüller space with respect to the Teichmüller distance.

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Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space II

Cited by 21 publications

References 20 publications

The asymptotic geometry of the Teichmüller metric

The asymptotic geometry of the Teichmüller metric

Geometry of the Gromov product: Geometry at infinity of Teichmüller space

Unification of extremal length geometry on Teichmüller space via intersection number

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