Rigidity of Teichmüller space

Eskin, Alex; Masur, Howard; Rafi, Kasra

doi:10.2140/gt.2018.22.4259

Cited by 7 publications

(8 citation statements)

References 27 publications

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“…To obtain these results, we show that the Weil-Petersson space admits a natural ternary operation, well defined up to bounded distance, which gives it the structure of a coarse median space, as defined in [Bo1]. We remark that related results for the Teichmüller metric have been proven elsewhere, see [EMR1,Bo7,EMR2].…”

Section: Introductionmentioning

confidence: 61%

Large-scale rank and rigidity of the Weil–Petersson metric

Bowditch

2020

Groups Geom. Dyn.

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Section: Introductionmentioning

confidence: 61%

Large-scale rank and rigidity of the Weil–Petersson metric

Bowditch

2020

Groups Geom. Dyn.

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“…While the delails are (significantly) different, related arguments can be made to work for quasi-isometries of T and W, giving the rigidity results for these spaces [Bo5,Bo7]. We remark that the rigidity of T is independently proven in [EMR2] using quite different arguments of coarse differentiation.…”

Section: Definitionmentioning

confidence: 96%

Notes on coarse median spaces

Bowditch¹

2019

Beyond Hyperbolicity

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“…Given an appropriate (case-specific) replacement for the notation of "affine mapping," one can formulate notions of "differentiation" in many settings that do not necessarily involve linear spaces; examples of such "qualitative" metric differentiation results include [71,46,16,73,45,51,18,18,19,20]. Corresponding results about quantitative differentiation, which lead to refined (often quite subtle and important) rigidity results can be found in [7,43,67,59,21,52,74,75,22,53,17,29,62,30,31,24,23,2,54,32]. Due to the prominence of this topic and the fact that many of the quoted results are probably not sharp, it would be of interest to develop new methods to prove sharper quantitative differentiation results.…”

Section: Amentioning

confidence: 99%

“…where in the last step of (37) we used the upper bound on t that appears in (32). Our choice of p in (37) is p = C M (X)…”

Section: The Lipschitz Constant Of Heat Evolutesmentioning

confidence: 99%

Heat flow and quantitative differentiation

Hytönen

2019

J. Eur. Math. Soc.

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For every Banach space (Y, · Y ) that admits an equivalent uniformly convex norm we prove that there exists c = c(Y ) ∈ (0, ∞) with the following property. Suppose that n ∈ N and that X is an n-dimensional normed space with unit ball BX . Then for every 1-Lipschitz function f : BX → Y and for every ε ∈ (0, 1/2] there exists a radius r exp(−1/ε cn ), a point x ∈ BX with x + rBX ⊆ BX , and an affine mapping Λ :εr for every y ∈ x+rBX. This is an improved bound for a fundamental quantitative differentiation problem that was formulated by Bates, Johnson, Lindenstrauss, Preiss and Schechtman (1999), and consequently it yields a new proof of Bourgain's discretization theorem (1987) for uniformly convex targets. The strategy of our proof is inspired by Bourgain's original approach to the discretization problem, which takes the affine mapping Λ to be the first order Taylor polynomial of a time-t Poisson evolute of an extension of f to all of X and argues that, under appropriate assumptions on f , there must exist a time t ∈ (0, ∞) at which Λ is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that Λ well-approximates f on a macroscopically large ball, in which case we show that for our argument to work one cannot use the Poisson semigroup. Nevertheless, our strategy does succeed with the Poisson semigroup replaced by the heat semigroup. As a crucial step of our proof, we establish a new uniformly convex-valued Littlewood-Paley-Stein G-function inequality for the heat semigroup; influential work of Martínez, Torrea and Xu (2006) obtained such an inequality for subordinated Poisson semigroups but left the important case of the heat semigroup open. As a byproduct, our proof also yields a new and simple approach to the classical Dorronsoro theorem (1985) even for real-valued functions.

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Rigidity of Teichmüller space

Abstract: We prove that the every quasi-isometry of Teichmüller space equipped with the Teichmüller metric is a bounded distance from an isometry of Teichmüller space. That is, Teichmüller space is quasi-isometrically rigid.

Cited by 7 publications

References 27 publications

Large-scale rank and rigidity of the Weil–Petersson metric

Large-scale rank and rigidity of the Weil–Petersson metric

Notes on coarse median spaces

Heat flow and quantitative differentiation

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