Pants decompositions and the Weil-Petersson metric

Since notions of coarse geometry and quasi-isometries were first introduced by M. Gromov, many studies of geometry have been renovated with its rough perspective. In this note we give an expository account of results of [Br] and of joint work of the author with Benson Farb [BF] that apply such a coarse point of view to the Weil-Petersson metric on Teichmüller space. A natural graph of pants decompositions of surfaces, introduced by A. Hatcher and W. Thurston, provides an organizing combinatorial structure for the Weil-Petersson metric. The coarse structure of this graph provides a new tool in the geometric study of Teichmüller spaces. We briefly introduce the two objects to be compared.