Speaker
Description
Our starting points consist of the simultaneous uniformization theorem for surface groups and the mating construction for polynomials. We shall 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 characterize the combinations of polynomials and Fuchsian genus zero orbifold groups as explicit algebraic functions. This allows us to embed the 'product' of Teichmüller spaces of genus zero orbifolds and parameter spaces of polynomials in a larger ambient space of algebraic correspondences. We conclude by describing the analog of the Bers slice in this context. and establish the pre-compactness of the Bers slice. This is the analog of the Bers' compactness theorem in classical Teichmüller theory. Joint work with Yusheng Luo and Sabyasachi Mukherjee.
