Le Spectre Marque des Longueurs des Surfaces a Courbure Negative

Otal, Jean-Pierre

doi:10.2307/1971511

Cited by 209 publications

(209 citation statements)

References 9 publications

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“…Proposition 4.16, Proposition 4.17 and Lemma 4.18 are easy modifications of J.-P. Otal's results in [Ota1,Proposition 6,7 and Lemma 8]. We only emphasize the differences with the non singular case.…”

Section: Negatively Curved Surfaces With Same Marked Length Spectrummentioning

confidence: 79%

“…We end this subsection by proving the implications (2) ⇒ (3) and (3) ⇒ (2) of Theorem B, inspired by [Ota1,Ota2]. Proof.…”

Section: Definition 42 the Intersection Number Of µ And γ Is I(µ γmentioning

confidence: 90%

“…The aim of this final part is to prove that (2)+(3) implies (4) in Theorem C. We will follow closely J.-P. Otal's proof in [Ota1], emphasizing mainly the points where the proofs are different.…”

Section: Negatively Curved Surfaces With Same Marked Length Spectrummentioning

confidence: 97%

“…Hence we will only have to prove in what follows that (2) and (3) are equivalent (subsection 4.1) and that (2) and (3) imply (4) (subsection 4.4). Note that if there are no singularities, (3) implies (4) is due to J.-P. Otal [Ota1].…”

Section: Marked Length Spectrum Rigidity For Negatively Curved Surfacmentioning

confidence: 99%

“…In his seminal work (see [Sul1,Sul2]) on the ergodic theory at infinity of discrete isometry groups of H n R , D. Sullivan [Sul2] extended Mostow's rigidity result to real hyperbolic n-manifolds whose volume grows slower than the volume able" extensions of J.-P. Otal's arguments [Ota1], in particular concerning the Liouville current. We conclude as in [Ota1]. Note that Theorem C also holds for other singular negatively curved Riemannian metrics with geodesic flow having singularities on a measure 0 subset, in a sense we will not make precise.…”

Section: Introductionmentioning

confidence: 99%

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On the rigidity of discrete isometry groups of negatively curved spaces

Hersonsky¹,

Paulin²

1997

Commentarii Mathematici Helvetici

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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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Section: Negatively Curved Surfaces With Same Marked Length Spectrummentioning

confidence: 79%

“…We end this subsection by proving the implications (2) ⇒ (3) and (3) ⇒ (2) of Theorem B, inspired by [Ota1,Ota2]. Proof.…”

Section: Definition 42 the Intersection Number Of µ And γ Is I(µ γmentioning

confidence: 90%

Section: Negatively Curved Surfaces With Same Marked Length Spectrummentioning

confidence: 97%

Section: Marked Length Spectrum Rigidity For Negatively Curved Surfacmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the rigidity of discrete isometry groups of negatively curved spaces

Hersonsky¹,

Paulin²

1997

Commentarii Mathematici Helvetici

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Rigidity Theorems in Riemannian Geometry

Croke

2004

Geometric Methods in Inverse Problems and PDE Control

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Flexibility of Group Actions on the Circle

Mj¹,

Koberda²,

Kim³

2019

Lecture Notes in Mathematics

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In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if L is a nontrivial limit group then the nonlinear representation variety HompL, Homeo`pS 1 qq contains uncountably many semi-conjugacy classes of faithful actions on S 1 with pairwise disjoint rotation spectra (except for 0) such that each representation lifts to R. For the case of most Fuchsian groups L, we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface group admits an action on S 1 that is never semi-conjugate to any action that factors through a finite-dimensional connected Lie subgroup in Homeo`pS 1 q. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate faithful actions on S 1 . In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of PSL 2 pRq which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of Z-valued subadditive defect-one quasimorphisms on these groups would follow as a corollary. We also give a mostly self-contained reconciliation of the various notions of semi-conjugacy in the extant literature by showing that they are all equivalent.

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Le Spectre Marque des Longueurs des Surfaces a Courbure Negative

Cited by 209 publications

References 9 publications

On the rigidity of discrete isometry groups of negatively curved spaces

On the rigidity of discrete isometry groups of negatively curved spaces

Rigidity Theorems in Riemannian Geometry

Flexibility of Group Actions on the Circle

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