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Some metrics on Teichmüller spaces of surfaces of infinite type

Authors: Lixin Liu and Athanase Papadopoulos
Journal: Trans. Amer. Math. Soc. 363 (2011), 4109-4134
MSC (2000): Primary 32G15, 30F30, 30F60
Published electronically: March 23, 2011
MathSciNet review: 2792982
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Abstract: Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topologically infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or conformal), on the choice of a distance function on Teichmüller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichmüller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichmüller theory of surfaces of infinite topological type that do not appear in the setting of the Teichmüller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichmüller spaces associated to a given surface of topologically infinite type.

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Additional Information

Lixin Liu
Affiliation: Department of Mathematics, Sun Yat-sen (Zongshan) University, 510275, Guangzhou, People’s Republic of China

Athanase Papadopoulos
Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France

Keywords: Teichmüller space, infinite-type surface, Teichmüller metric, quasiconformal metric, length spectrum metric, bi-Lipschitz metric
Received by editor(s): June 23, 2008
Received by editor(s) in revised form: March 16, 2009, and April 18, 2009
Published electronically: March 23, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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