The aim of this paper is to develop the theory of a compactification of Teichmüller space given by F. Gardiner and H. Masur, which we call the Gardiner-Masur compactification of the Teichmüller space. We first develop the general theory of the Gardiner-Masur compactification. Secondly, we will investigate the asymptotic behaviors of Teichmüller geodesic rays under the Gardiner-Masur embedding. In particular, we will observe that the projective class of a rational measured foliation G can not be an accumulation point of every Teichmüller geodesic ray under the Gardiner-Masur embedding, when the support of G consists of at least two simple closed curves.

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We prove that the bijective correspondence between the space of bounded measured laminations ML b (H) and the universal Teichmüller space T (H) given by λ → E λ | S 1 is a homeomorphism for the uniform weak* topology on ML b (H) and the Teichmüller topology on T (H), where E λ is an earthquake with an earthquake measure λ. A corollary is that earthquakes with discrete earthquake measures are dense in T (H). We also establish infinitesimal versions of the above results. We are particularly interested in the geometrically infinite hyperbolic Riemann surfaces, for example, the hyperbolic plane H, an infinite genus surface, a surface with an interval of ideal boundary points. All these surfaces have infinite hyperbolic area.

In this paper, we give a framework for the study of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur. There is a natural compactification using extremal length geometry introduced by Gardiner and Masur. The compactification is realized in a certain projective space. We develop the extremal length geometry in the cone which is defined as the inverse image of the compactification via the quotient mapping. The compactification is identified with a subset of the cone by taking an appropriate lift. The cone contains canonically the space of measured foliations in the boundary. We first extend the geometric intersection number on the space of measured foliations to the cone, and observe that the restriction of the intersection number to Teichmüller space is represented by an explicit formula in terms of the Gromov product with respect to the Teichmüller distance. From this observation, we deduce that the Gromov product extends continuously to the compactification. As an application, we obtain an alternative approach to a characterization of the isometry group of Teichmüller space. We also obtain a new realization of Teichmüller space, a hyperboloid model of Teichmüller space with respect to the Teichmüller distance.