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

TL;DR

This paper demonstrates that the Teichmüller and length-spectra metrics are almost isometric on moduli space, providing a new understanding of their geometric relationship through a simple model based on the complex of curves.

Contribution

It establishes an almost isometry between the length-spectra metric and a cone model over the complex of curves, extending Farb-Masur's theorem to moduli space.

Findings

Length-spectra metric is almost isometric to a cone model over the complex of curves.

Teichmüller and length-spectra metrics are almost isometric on moduli space.

They are not quasi-isometric on Teichmüller 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.