Some metrics on Teichmüller spaces of surfaces of infinite type

Liu, Lixin; Papadopoulos, Athanase

doi:10.1090/S0002-9947-2011-05090-7

Some metrics on Teichmüller spaces of surfaces of infinite type
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by Lixin Liu and Athanase Papadopoulos PDF

Trans. Amer. Math. Soc. 363 (2011), 4109-4134 Request permission

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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