The decorated Teichmüller space of punctured surfaces

Abstract

A principal ℝ ⁵₊ -bundle over the usual Teichmüller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several coordinatizations of the total space of the bundle are developed. There is furthermore a natural cell-decomposition of the bundle. Finally, we compute the coordinate action of the mapping class group on the total space; the total space is found to have a rich (equivariant) geometric structure. We sketch some connections with arithmetic groups, diophantine approximations, and certain problems in plane euclidean geometry. Furthermore, these investigations lead to an explicit scheme of integration over the moduli spaces, and to the construction of a “universal Teichmüller space,” which we hope will provide a formalism for understanding some connections between the Teichmüller theory, the KP hierarchy and the Virasoro algebra. These latter applications are pursued elsewhere.