Coverings, algebraic functions, and conformal module of braids

We consider coverings of open Riemann surfaces
(ramified or unramified) and liftings of coverings to embeddings
into holomorphic disc bundles. Coverings and liftings to smooth embeddings can be described in algebraic terms. Liftings to holomorphic embeddings carry a conformal aspect --- one of
the manifestations of the well-known difference between the symplectic and the complex world.
There are restrictions on the conformal classes of base manifolds for which a given covering lifts to some holomorphic embedding. (But there are always some elements of the Teichm\uller space for which this is so.) A restriction can be given in terms of assigning to each closed braid the conformal module of certain annulus (in the sense of Ahlfors).
All but some exceptional closed braids have finite conformal module. At least for closed three-braids the conformal module is related to entropy. Applications e.g. to the theory of algebraic functions on Riemann surfaces will be given.