There are two frameworks for mating Kleinian groups with rational maps on the Riemann sphere: an algebraic correspondence framework due to Bullett-Penrose-Lomonaco and an orbit equivalence mating framework using Bowen-Series maps. The latter is analogous to the Douady-Hubbard theory for polynomial mating. We will discuss how these two frameworks can be unified and generalized. As a consequence, we will construct holomorphic correspondences that are matings of hyperbolic orbifold groups (including Hecke groups) with Blaschke products. Time permitting, we will introduce an analog of a Bers slice of the above orbifolds in the algebraic parameter space of correspondences.

Based on joint work with Mahan Mj.