Infinite-dimensional Teichm{ü}ller spaces

Abstract

In this paper, the Teichmüller spaces of surfaces appear from two points of views: the conformal category and the hyperbolic category. In contrast to the case of surfaces of topologically finite type, the Teichmüller spaces associated to surfaces of topologically infinite type depend on the choice of a base structure. In the setting of surfaces of infinite type, the Teichmüller spaces can be endowed with different distance functions such as the length-spectrum distance, the bi-Lipschitz distance, the Fenchel-Nielsen distance, the Teichmüller distance and there are other distance functions. Unlike the case of surfaces of topologically finite type, these distance functions are not equivalent.We introduce the finitely supported Teichmüller space T f s H 0 associated to a base hyperbolic structure H 0 on a surface Σ, provide its characterization by Fenchel-Nielsen coordinates and study its relation to the other Teichmüller spaces. This paper also involves a study of the Teichmüller space T 0 ls H 0 of asymptotically isometric hyperbolic structures and its Fenchel-Nielsen parameterization. We show that T f s H 0 is dense in T 0 ls H 0 , where both spaces are considered to be subspaces of the length-spectrum Teichmüller space T ls H 0 . Another result we present here is that asymptotically length-spectrum bounded Teichmüller space A T ls H 0 is contractible. We also prove that if the base surface admits short curves then the orbit of every finitely supported hyperbolic surface is non-discrete under the action of the finitely supported mapping class group MCG f s Σ .