Extremal length geometry

Abstract

This is a survey of the development of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur’s works. We shall discuss intersection number in extremal length geometry, and we shall view a canonical compactification in extremal length geometry with respect to intersection number. This is an analogy with the Thurston compactification in hyperbolic geometry. We will also obtain a hyperboloid model of Teichmüller space in extremal length geometry. This is an analogy with a realization of Teichmüller space into the space of the geodesic currents observed by F. Bonahon.