The Bers-Greenberg Theorem and the Maskit Embedding for Teichm\"{u}ller spaces

Abstract

The Bers-Greenberg theorem tells that the Teichm\"{u}ller space of a Riemann surface with branch points (orbifold) depends only on the genus and the number of special points, but not on the particular ramification values. On the other hand, the Maskit embedding provides a mapping from the Teichm\"{u}ller space of an orbifold, into the product of one dimensional Teichm\"{u}ller spaces. In this paper we prove that there is a set of isomorphisms between one dimensional Teichm\"{u}ller spaces that, when restricted to the image of the Teichm\"{u}ller space of an orbifold under the Maskit embedding, provides the Bers-Greenberg isomorphism.