Published online by Cambridge University Press: 05 November 2011
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.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.