Combination theorems, Bers slices, and holomorphic correspondences

Abstract

Our starting points consist of the simultaneous uniformization theorem for surface groups and the mating construction for polynomials. In part I of the talk, we describe a hybrid construction that simultaneously uniformizes a polynomial and a surface. We provide two constructions for some genus zero orbifolds and polynomials lying in the principal hyperbolic component:
1) For punctured spheres with possibly order 2 orbifold points using orbit equivalence
2) Generalizing (1) to orbifolds that have, in addition, an orbifold point of order > 2. This uses a factor dynamical system.
We conclude by describing the analog of the Bers slice in this context.
In the second part, we will characterize the combinations of polynomials and Fuchsian genus zero orbifold groups as explicit algebraic functions. This allows us to embed the 'product' of Teichm{\"u} spaces of genus zero orbifolds and parameter spaces of polynomials in a larger ambient space of algebraic correspondences.
We will discuss compactifications of such copies of Teichm{\"u}ller spaces in the space of correspondences, and end with a host of open questions.