Almost-isometry between Teichm\"{u}ller metric and length-spectra metric on moduli space

Abstract

We prove an analogue of Farb-Masur's theorem that the length-spectra metric on moduli space is "almost isometric" to a simple model $\mathcal {V}(S)$ which is induced by the cone metric over the complex of curves. As an application, we know that the Teichm\"{u}ller metric and the length-spectra metric are "almost isometric" on moduli space, while they are not even quasi-isometric on Teichm\"{u}ller space.