On asymptotic Teichmüller space

Fletcher, Alastair

doi:10.1090/s0002-9947-09-04944-7

Cited by 5 publications

(9 citation statements)

References 18 publications

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“…In particular, any two infinite-dimensional Teichmüller spaces are locally bi-Lipschitz equivalent under the Teichmüller metric. A similar result was proved in [8] for asymptotic Teichmüller spaces.…”

Section: Carathéodory and Kobayashi Metrics On At: Continuedsupporting

confidence: 78%

“…Precisely, if C AT < 1 9 , then C AT τ 9C AT /(1 − 9C AT ), and ifτ τ 0 , then C AT τ 9(1 +τ 0 )C AT . Before giving the proof of theorems 9.1 and 9.2, we state some recent results obtained by Fletcher [7,8] which are very relevant to the present paper. Fletcher [7] proved that, on any infinite-dimensional Teichmüller space, the Teichmüller metric is locally bi-Lipschitz equivalent to the l ∞ metric on the space of bounded sequences.…”

Section: Carathéodory and Kobayashi Metrics On At: Continuedmentioning

confidence: 99%

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The asymptotic Teichmüller space and the asymptotic Grunsky map

Shen¹

2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Section: Carathéodory and Kobayashi Metrics On At: Continuedsupporting

confidence: 78%

Section: Carathéodory and Kobayashi Metrics On At: Continuedmentioning

confidence: 99%

The asymptotic Teichmüller space and the asymptotic Grunsky map

Shen¹

2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…and |t n (Y k ) − t n (X)| → 0 as n → ∞ for each fixed k. Since the normalized Fenchel-Nielsen map is homeomorphism for the length spectrum distance, the above conditions imply that the limit Y satisfies (6) and (7).…”

Section: Proof Of Theorem 2 Consider a Sequence Of Marked Surfacesmentioning

confidence: 96%

Fenchel-Nielsen coordinates for asymptotically conformal deformations

Šaric¹

2016

Ann. Acad. Sci. Fenn. Math.

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Abstract. Let X be an infinite hyperbolic surface endowed with an upper bounded geodesic pants decomposition. Alessandrini, Liu, Papadopoulos, Su and Sun [2,3] parametrized the quasiconformal Teichmüller space T qc (X) and the length spectrum Teichmüller space T ls (X) using the Fenchel-Nielsen coordinates. A quasiconformal map f : X → Y is said to be asymptotically conformal if its Beltrami coefficient µ =∂f /∂f converges to zero at infinity. The space of all asymptotically conformal maps up to homotopy and post-composition by conformal maps is called "little" Teichmüller space T 0 (X). We find a parametrization of T 0 (X) using the Fenchel-Nielsen coordinates and a parametrization of the closure T 0 (X) of T 0 (X) in the length spectrum metric. We also prove that the quotients AT (X) = T qc (X)/T 0 (X), T ls (X)/T qc (X) and T ls (X)/T 0 (X) are contractible in the Teichmüller metric and the length spectrum metric, respectively. Finally, we show that the Wolpert's lemma on the lengths of simple closed geodesics under quasiconformal maps is not sharp.

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“…This is done using the Bers embedding theorem and employing techniques similar to those in [5]. Assume the results of Theorems 1.3 and 1.4 hold.…”

Section: 2mentioning

confidence: 99%

The Moduli Space of Riemann Surfaces of Large Genus

2013

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Abstract. Let M g,ǫ be the ǫ-thick part of the moduli space M g of closed genus g surfaces. In this article, we show that the number of balls of radius r needed to cover M g,ǫ is bounded below by (c 1 g) 2g and bounded above by (c 2 g) 2g , where the constants c 1 , c 2 depend only on ǫ and r, and in particular not on g. Using this counting result we prove that there are Riemann surfaces of arbitrarily large injectivity radius that are not close (in the Teichmüller metric) to a finite cover of a fixed closed Riemann surface. This result illustrates the sharpness of the Ehrenpreis conjecture.

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On asymptotic Teichmüller space

Abstract: Abstract. In this article we prove that for any hyperbolic Riemann surface M of infinite analytic type, the little Bers space Q 0 (M ) is isomorphic to c 0 .

Cited by 5 publications

References 18 publications

The asymptotic Teichmüller space and the asymptotic Grunsky map

The asymptotic Teichmüller space and the asymptotic Grunsky map

Fenchel-Nielsen coordinates for asymptotically conformal deformations

The Moduli Space of Riemann Surfaces of Large Genus

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