Almost-isometry between Teichmüller metric and length-spectrum metric on moduli space

摘要

We prove an analogue of Farb-Masur's theorem that the length-spectrum metric on moduli space is 'almost isometric' to a simple model (S) which is induced by the cone metric over the complex of curves. As an application, we know that the Teichmüller metric and the length-spectrum metric are 'almost isometric' on moduli space, while they are not even quasi-isometric on Teichmüller space.