A combination theorem for covering correspondences and an application to mating polynomial maps with Kleinian groups

The simplest version of the Maskit-Klein combination theorems concerns the action of a free product of two finite subgroups of $PSL(2,{\mathbb C})$ on the Riemann sphere $\hat{\mathbb C}$, when these subgroups have fundamental domains whose interiors together cover $\hat{\mathbb C}$. We prove an analogous combination theorem for covering correspondences of rational maps, making use of Douady and Hubbard's Straightening Theorem for polynomial-like maps to describe the structure of the limit sets. We apply our theorem to construct holomorphic correspondences which are matings of polynomial maps with Hecke groups $C_p*C_q$, and we show how it may also be applied to the analysis of separable correspondences.