Combining rational maps and Kleinian groups via orbit equivalence

We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that Kleinian surface groups. show only torsion-free Fuchsian groups can be thus mated are punctured sphere describe class maps topologically equivalent to call these higher Bowen-Series maps. The existence this ensures Teichm\"uller space matings has one component corresponding and also that, unlike in dimensions, topological rigidity fails acting on circle. classify collection Bers boundary mateable our framework.