The geometry of Teichm�ller space via geodesic currents

Bonahon, Francis

doi:10.1007/bf01393996

Cited by 277 publications

(498 citation statements)

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“…The map m −→ L m assigning to each hyperbolic metric m on S its Liouville current L m (see [Bo1]) induces a proper topological embedding of the Teichmüller space…”

Section: Measured Geodesic Laminations and T T (X)mentioning

confidence: 99%

“…In fact, ML(X) gets identified with such currents (see section 3 of [Bo1]). Consequently, the light cone comprising geodesic currents of self-intersection zero is homeomorphic to ML(X).…”

Section: Measured Geodesic Laminations and T T (X)mentioning

confidence: 99%

“…(i) By the work of F. Bonahon [Bo1], [Bo2], the Thurston-compactified Teichmüller space is described in terms of the G-invariant geodesic currents on the unit disc. Then T T (p) is defined by sending any G-invariant geodesic current on the universal covering disc ∆ to the same current considered as an H-invariant object.…”

Section: Such That T T (P) Is the Continuous Extension Of The Holomormentioning

confidence: 99%

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Thurston boundary of Teichmüller spaces and the commensurability modular group

Biswas

Mitra

Nag

1999

Conform. Geom. Dyn.

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Abstract. If p : Y → X is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to p, from the Teichmüller space T (X), for X, to T (Y ) actually extends to an embedding between the Thurston compactification of the two Teichmüller spaces. Using this result, an inductive limit of Thurston compactified Teichmüller spaces has been constructed, where the index for the inductive limit runs over all possible finite unramified coverings of a fixed compact oriented surface of genus at least two. This inductive limit contains the inductive limit of Teichmüller spaces, constructed by I. Biswas, S. Nag and D. Sullivan, Determinant bundles, Quillen metrics and Mumford isomorphisms over the Universal Commensurability Teichmüller Space, Acta Mathematica, 176 (1996), 145-169, as a subset. The universal commensurability modular group, which was constructed in the above mentioned article, has a natural action on the inductive limit of Teichmüller spaces. It is proved here that this action of the universal commensurability modular group extends continuously to the inductive limit of Thurston compactified Teichmüller spaces.

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“…The map m −→ L m assigning to each hyperbolic metric m on S its Liouville current L m (see [Bo1]) induces a proper topological embedding of the Teichmüller space…”

Section: Measured Geodesic Laminations and T T (X)mentioning

confidence: 99%

“…In fact, ML(X) gets identified with such currents (see section 3 of [Bo1]). Consequently, the light cone comprising geodesic currents of self-intersection zero is homeomorphic to ML(X).…”

Section: Measured Geodesic Laminations and T T (X)mentioning

confidence: 99%

Section: Such That T T (P) Is the Continuous Extension Of The Holomormentioning

confidence: 99%

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Thurston boundary of Teichmüller spaces and the commensurability modular group

Biswas

Mitra

Nag

1999

Conform. Geom. Dyn.

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“…We are first going to recall the definition of a geodesic current, developped by K. Sigmund, D. Sullivan and F. Bonahon [Bon1,Bon2,Bon3], to which we refer for basic properties and historical remarks.…”

Section: The Möbius Current Of a Cat(−1) Surfacementioning

confidence: 99%

On the rigidity of discrete isometry groups of negatively curved spaces

Hersonsky¹,

Paulin²

1997

Comment. Math. Helv.

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Abstract. We prove an ergodic rigidity theorem for discrete isometry groups of CAT(−1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2-polyhedra, hyperbolic Bruhat-Tits buildings and rank one symmetric spaces. We prove that two negatively curved Riemannian metrics, with conical singularities of angles at least 2π, on a closed surface, with boundary map absolutely continuous with respect to the Patterson-Sullivan measures, are isometric. For that, we generalize J.-P. Otal's result to prove that a negatively curved Riemannian metric, with conical singularities of angles at least 2π, on a closed surface, is determined, up to isometry, by its marked length spectrum.Mathematics Subject Classification (1991). 57S30, 53C45, 20H10, 51M10, 51E24.

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“…The following result may be considered as an analogous for the Möbius group of Propositions 1.1.1, 1.2.2, or 1.2.3. We refer to [19] for its proof (see also Exercise 1.3.11). Proposition 1.3.6.…”

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confidence: 99%

Groups of Circle Diffeomorphisms

Navas¹

2011

152

171

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The original version (in Spanish) of this text was prepared for a mini-course in Antofagasta, Chile. Subsequently, enlarged and revised versions were published in the series Monografías del IMCA (Perú) and Ensaios Matemáticos (Brasil). This translation arose from the necessity of making this text accessible to a larger audience. I would like to thank Juan Rivera-Letelier for his invitation to the II Workshop on Dynamical Systems ( 2001), for which the original version was prepared, and Roger Metzger for his invitation to IMCA (2006), where part of this material was presented. I would also like to thank both Étienne Ghys and Maria Eulália Vares for motivating me and allowing me to publish the revised Spanish version in Brasil, as well as all the people who strongly encouraged me to conclude this English version.This work was partially supported by CONICYT and PBCT via the Research Network on Low Dimensional Dynamics. I would also to acknowledge the support of both the UMPA Department of the École Normale Supérieure de Lyon, where the idea of writing this text was born during my PhD thesis, and the Institut des Hautes Études Scientifiques, where the original notes started taking its definite form while I benefited from a one-year postdoctoral position.This work owes a lot to many of my colleges. Several remarks spread throughout the text and the content of some examples and exercises were born during fruitful and quite stimulating discussions. It is then a pleasure to thank Sylvain Crovisier (Proposition 4.2.25),

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The geometry of Teichm�ller space via geodesic currents

Cited by 277 publications

References 14 publications

Thurston boundary of Teichmüller spaces and the commensurability modular group

Thurston boundary of Teichmüller spaces and the commensurability modular group

On the rigidity of discrete isometry groups of negatively curved spaces

Groups of Circle Diffeomorphisms

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