Unification of extremal length geometry on Teichmüller space via intersection number

Abstract

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.

Geometry of the Gromov product: Geometry at infinity of Teichmüller space

“…Theorem 4 and Proposition 7 in [33]). The extremal length (36) also satisfies the following generalized Minsky inequality:…”

“…(5.6) in [33]). From (33) and Gardiner-Masur's work in [10], (36) coincides with the original extremal length when a ∈ MF .…”

“…From Proposition 1 of [33], Ψ GM : T → C GM extends to an injective continuous mapping on cl GM (T ).…”

“…In this section, we recall the unification of extremal length geometry via intersection number developed in [33].…”

“…In [33], the author established the following unification of extremal length geometry via the intersection number. Theorem 6.1 (Theorem 1.1 in [33]).…”