The Weil–Petersson geometry of the five-times punctured sphere

Summary

Abstract

We give a new proof that the completion of the Weil–Petersson metric on Teichmüller space is Gromov-hyperbolic if the surface is a five-times punctured sphere or a twice-punctured torus. Our methods make use of the synthetic geometry of the Weil–Petersson metric.

Introduction

The large scale geometry of Teichmüller space has been a very important tool in different aspects of the theory of hyperbolic 3-manifolds. Within this context, a natural question to ask is whether Teichmüller space, with a given metric, is hyperbolic in the sense of Gromov (or Gromov hyperbolic, for short). In general, the answer is negative. In their paper [BF01], the authors prove the following: if ∑ is a surface of genus g and with p punctures, with 3g-3+p > 2, then the Teichmüller space of ∑, endowed with the Weil–Petersson metric, is not Gromov hyperbolic. In the case when 3g-3+p=2 (that is, when the surface is a sphere with five punctures or a twice-punctured torus) they show that the Weil–Petersson Teichmüller space is Gromov hyperbolic. The proof makes reference to very deep results by Masur and Minsky [MM99] on the Gromov hyperbolicity of the curve complex. We remark that Behrstock [Beh05] has also used the geometric structure of the curve complex to give a new proof of the hyperbolicity of the Weil–Petersson metric for these “low-complexity” cases.