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

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 Teichmüller metric and the length-spectra metric are "almost isometric" on moduli space, while they are not even quasi-isometric on Teichmüller space.

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