A Thurston boundary for infinite-dimensional Teichmüller spaces

Bonahon, Francis; Šarić, Dragomir

doi:10.1007/s00208-021-02148-z

Cited by 6 publications

(7 citation statements)

References 25 publications

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“…When X is a compact surface the space of geodesic currents G(X) is equipped with the standard weak* topology for which the Liouville map is an embedding onto its image (see Bonahon [3]). Bonahon and the second author [4] introduced the uniform weak* topology on the space of geodesic currents in order to introduce a Thurston boundary to Teichmüller spaces of arbitrary Riemann surfaces. This is a simplification of the topology that was introduced by the second author [10].…”

Section: The Liouville Map and Uniform Hölder Topologymentioning

confidence: 99%

“…The second author [10], and more recently, Bonahon and the second author [4] introduced the Thurston boundary to Teichmüller spaces of arbitrary conformally hyperbolic Riemann surfaces. In the case of an infinite area conformally hyperbolic Riemann surface, we need a uniform bound on deformations which was automatic for closed Riemann surfaces.…”

mentioning

confidence: 99%

“…More recently, Bonahon and the second author [4] introduced a simpler family of seminorms that describes the same topology on the space of bounded geodesic currents G b (X) for an arbitrary conformally hyperbolic Riemann surface X. Let ζ : G( X) → R be a continuous function with compact support.…”

mentioning

confidence: 99%

“…where the supremum is over all isometries γ of X. Using the topology on G b (X) induced by the family of semi-norms { • ζ }, Bonahon and the second author [4] proved that the Liuoville map L : T(X) → G b (X) is an embedding onto its image and that the boundary of the projectivization of L(T(X)) is exactly equal to the space of projective bounded measured laminations-a Thurston boundary to T(X).…”

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confidence: 99%

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On complex extension of the Liouville map

Dong¹,

Šarić²

2021

Preprint

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The Liouville map assigns to each point in the Teichmüller space a positive Radon measure on the space of geodesics of the universal covering of the base Riemann surface. This construction which was introduced by Bonahon is valid for both finite and infinite Riemann surfaces. Bonahon and Sözen proved that the Liouville map is differentiable for closed Riemann surfaces and the second author extended this result to all other Riemann surfaces. Otal proved that the Liouville map is real analytic using an idea from the geometric analysis. The purpose of this note is to give another proof of Otal's result using a complex analysis approach.

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Section: The Liouville Map and Uniform Hölder Topologymentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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On complex extension of the Liouville map

Dong¹,

Šarić²

2021

Preprint

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“…Geodesic currents also play a key step in the proof of rigidity of the marked length spectrum for metrics, via an argument by Otal [47, Théorème 2]. Finally, they provide a boundary of the Teichmüller space, in both the compact [8, Proposition 17] and noncompact [11, Theorem 2] cases.…”

Section: Introductionmentioning

confidence: 99%

From curves to currents

Martínez-Granado

Thurston

2021

Forum of Mathematics, Sigma

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Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

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The Heights Theorem for infinite Riemann surfaces

Šarić

2022

Geom Dedicata

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A Thurston boundary for infinite-dimensional Teichmüller spaces

Cited by 6 publications

References 25 publications

On complex extension of the Liouville map

On complex extension of the Liouville map

From curves to currents

The Heights Theorem for infinite Riemann surfaces

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