Some metrics on Teichmüller spaces of surfaces of infinite type

Abstract

Unlike the case of surfaces of topologically finite type, there are several different Teichmuller spaces that are associated to a surface of topo- logical infinite type. These Teichmuller 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 confor- mal), on the choice of a distance function on Teichmuller 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 Teichmuller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichmuller theory of surfaces of infinite topological type that do not appear in the setting the Teichmuller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichmuller spaces associated to a given surface of topological infinite type. AMS Mathematics Subject Classification: 32G15 ; 30F30 ; 30F60.