Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms

A quadratic differential on a Riemann surface M determines certain data: the genus o f M; the orders o f zeros and poles; and the orientability o f the horizontal foliation. In this note we determine which collections o f data can be realized by quadratic differentials with finite area. A pseudo-Anosov diffeomorphism of M also determines certain topological data: the genus of the M; the types of the singularities and the orientability of the stable foliation. As a corollary to our result on quadratic differentials we determine which topological data can be realized by pseudo-Anosov diffeomorphisms on oriented surfaces. Let X be a closed Riemann surface o f genus g with a system of holomorphic coordinate charts U,. , h,. . This means that U, is a covering of X by open sets; h,. is a homeomorph i sm of U,. to an open set in the complex plane and hu ~ h; ~ is conformal whenever defined. Let q be a positive integer. A meromorphic q-differential 05 on X is a set o f meromorphic function elements 05, , in the local parameters zv = h,. (p) for which the t ransformat ion law