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.

Appendix: A proper geodesic metric space without extendable Gromov product

This section is devoted to giving a geodesic metric space to which the Gromov product does not extend on the horofunction boundary. The following example is given by Cormac Walsh (cf. [37]). Notice that the Gardiner–Masur compactfication coincides with the horofunction compactification with respect to the Teichmüller distance (cf. [18]).

Let $C_{n}$ be the frame $\partial ([-n,n]\times [0,n])$ with the standard Euclidean metric. We construct a space $X$ by gluing each frame $C_{n}$ to $\mathbb {R}$ along the bottom edge $[-n,n]\times \{0\}$ of $C_{n}$ and the interval $[-n,n]$ of $\mathbb {R}$ isometrically. The space $X$ is a proper geodesic space (cf. Fig. 3).

Fig. 3

The metric space $X$

Let $b_{0}$, $x^{1}_{n},y^{1}_{n},x^{2}_{n}$ and $y^{2}_{n}$ be points in $X$ corresponding to $0\in \mathbb {R}$, $(-n,0)$, $(-n,n)$, $(n,0)$ and $(n,n)$ in $C_{n}$ respectively. We consider $b_{0}$ as the basepoint of $X$. Then, one can see that for $i=1,2$, $\{x^{i}_{n}\}_{n}$ and $\{y^{i}_{n}\}_{n}$ converges to the same Busemann point in the horofunction boundary of $X$ though $\{y^{i}_{n}\}_{n}$ is not an almost geodesic (cf. [33]). On the other hand, we see

$$\begin{aligned} \lim _{n\rightarrow \infty } \langle y^{1}_{n}\,|\,y^{2}_{n}\rangle _{b_{0}} =\lim _{n\rightarrow \infty }\frac{1}{2}(2n+2n-2n)=\infty \end{aligned}$$

while $\langle x^{1}_{n}\,|\,x^{2}_{n}\rangle _{b_{0}}=(n+n-2n)/2=0$ for all $n$.